You can replace the granulated sugar with confectioner's sugar, but due to that starch, you'll end up with an icing-like consistency. In fact, royal icing is just egg whites and confectioner's sugar.
There is actually a surprisingly large number of variables when it comes to beating egg whites, with the amount and type of sugar being just one of them. I'm not sure if this is going to be a complete list, but here's the basics of what you need to know for beating up to the "stiff peak" stage (and if you want soft peaks, then just don't beat for as long):
Age of eggs. Older eggs will whip faster and to a greater volume, but will be less stable (i.e. will deflate faster). Most people seem to find they get the best results with eggs that are about 3 days old. Very fresh eggs are difficult to whip, and very old eggs (more than 1-2 weeks) are just too unstable to work with.
Temperature. You want to separate the eggs when cold, because the whites are firmer and it will be easier for you to cleanly separate the yolks from the whites. But you always want to allow the whites themselves to come up to room temperature before you start beating them, for the exact opposite reason - the proteins are less firm, more stretchy, and will more readily accept air.
Impurities (especially fat). Make sure you don't have any impurities. Use a clean, dry mixing bowl and whisk. Don't let any of the yolk get in; even a trace amount will significantly decrease the amount of air you can incorporate, and if you happen to contaminate the whites this way, it's probably better just to start over.
Salt and acid both help egg whites to coagulate, which you might remember if you've ever had to poach them. That has a stabilizing effect which will allow you to whip the eggs longer and firmer without blowing a hole in the web of proteins, and will also help to prevent (or at least slow down) the foam/meringue from deflating afterward. You don't need more than a pinch of salt, and the traditional amount of acid is either 1/8 tsp cream of tartar (preferred) or 1/4 tsp lemon juice or vinegar. You'll want to start adding these early, as soon as you see the egg whites starting to foam.
Finally, sugar is critical to attain the stiffest peaks, but it is critical that you do not add it too early or too quickly! Sugar serves a similar role here as it does in baking; it dissolves in the water, absorbing it and drawing it away from the proteins, which improves stability a lot. But this particular type of stabilization is a double-edged sword; if you add it too early, it will interfere with the coagulation process and it will take forever to beat them to peaks. And if you dump in all the sugar at once, it will just deflate the foam you've worked so hard to produce (and possibly break it, requiring you to start all over again with new eggs).
In general, you want to add the sugar right after you see soft peaks beginning to form, and incorporate it very gradually while beating. If you want the final result to be soft peaks instead of stiff peaks (which you do, in this case), you can add the sugar slightly early, but err on the side of caution here. Stiffer peaks won't harm your pudding that much, but no peaks definitely will. I also suggest using superfine sugar, because the finer the granules, the quicker they will dissolve; but don't use icing sugar unless you want icing.
Finally, don't over-beat the whites. As I alluded to above in the salt/acid section, the beating process involves stretching the proteins and incorporating air; incorporate too much air and you'll just break the proteins, and then you'll have to throw the eggs out and start all over again. This is especially easy to do accidentally if you use cold eggs.
If you keep all of these things in mind then you can whip the eggs all the way to a nice stiff meringue; soft peaks should be a piece of cake.
Best Answer
That would be a difficult substitution, because egg whites and egg yolks have very different roles in baking. It is even worse that the recipe is already not a robust standard recipe, but a lowfat, low-sugar recipe, where all tolerances have probably been used up in engineering a texture which works with the current substitutions. It is doable, but it is not a matter of "one look and one can tell what will work out", you would have to go through a process of trial and error.
To kick off your recipe redoing, start with the eggs only. The closest results will be by obtained by using two whole eggs. This will certainly give you a different outcome than intended by the author, it will be wetter, and may bake differently. If you don't like it that way, for the following attempts, start reducing the applesauce in 10% steps. This will dial back on the wetness problem. If that doesn't work and the change in applesauce amount only moves you between a dry, crumbly texture and wet, soggy texture, you might have to dial back to using only one whole egg and adding one or two egg whites, doing half the substitution so to speak. Another improvement may be gained by adding more fat back in, for example in the form of using cocoa powder with a high fat content.
In the end, it also depends on your personal preferences. For some people, only the best texture will do, and then the task is very difficult. For others, any brownie is a good brownie, and you can just drop in those whole eggs and enjoy.