Eggs – How to convey an order for over medium eggs without runny yolks

Your recipe doesn't specify 55°C, and I'd be surprised if 5–8 minutes over barely simmering water only gets that hot. Indeed, checking for sources:

• McGee, in On Food and Cooking, says:

  When the temperature reaches 120°F/50°C, high enough to unfold some of the yolk proteins, the mix thickens, traps air more efficiently, and begins to expand. As the proteins continue to unfold and then bond to each other, the foam rises into fluffy mounds. The key to maximally light zabaglione is to stop the heating just when the foam teeters on the cusp between liquid and solid. Further cooking will produce a stiffer, denser, eventually tough sponge as the proteins over-coagulate

  so that leads to the conclusion that 55°C wasn't hot enough, as it hadn't yet reached "the cusp between liquid and solid".

• Hans-Dieter Belitz, Werner Grosch, and Peter Schieberle say in Food Chemistry:

  Egg yolk can be whipped into stable foam only at higher temperatures (optimum 72°C), the volume increasing about sixfold in the process. Above the critical temperature, the volume falls and the proteins coagulate. The protein coagulation is prevented by reducing the pH value, e.g., by the addition of acetic acid. This effect is used in the production of highly stable sauces

  so that'd imply you want 72°C, or maybe higher due to the acid (wine) present. As a side benefit, that'd also pasteurize the eggs.

In summary, I think you should have cooked it hotter.

I doubt it is going to firm much in the fridge, hopefully it is stable enough to not further liquify.

It really does depend on the recipe. Egg yolks have some protein but a lot of fat which adds flavor and makes baked goods more tender and moist. Whites are just protein which adds structure and has a drying effect.

If the particular recipe is depending on the eggs for the binding power of their protein you may not get enough by substituting just yolks. Remember that the volume of the yolks is less than whole eggs so you may see better success substituting two whole eggs with three yolks.

Some cakes will handle the reduced protein. White and yellow cakes will be ok. Pound cakes will also be fine and in fact some recipes call for only yolks. Expect your cake to be a little denser, softer, and tastier. Any kind of sponge or angel food cake won't work because they rely wholly on the protein from the whites.

Soft, rich breads will also benefit from the added fat. The breads will derive their structure from the gluten in their flour. Cinnamon rolls and rich doughnuts come to mind. A local doughnut shop is famous for the yellow color which comes from using fresh yolks.

Yolks can be used in many recipes that call for whole eggs. Look for recipes that will benefit from all the added fat. Anything custard-like, where the eggs are mixed with milk, will work very well- even better than with whole eggs in fact. Examples are custard, flan, puddings, french toast, etc. Given the season you might consider a custard-based ice cream.