As you noticed, white chocolate doesn't have the same fat or solid makeup as regular chocolate, and in baking that will make a huge difference. The proportions of starch/binder (flour) to fat (butter & chocolate) to protein (eggs) are what make the difference between cake, brownie, fudge and everything in between.
White chocolate tends to melt faster than regular chocolate, so the baking time will be shorter, and there'll probably need to be extra flour & butter to make up the difference.
Go ahead and google for a basic recipe & then you should be able to make most of the same add-ins (nuts, extracts/flavorings) as any regular brownie recipe.
I would warn you to stay away from the "marble brownie" recipes though - they're really popular, but these are not two batters that play well with each other.
EDIT: Regular chocolate varies broadly in its fat makeup - different countries have diff regulations on what types of (natural) fats and (artificial) emulsifiers can be added to chocolate and chocolate-derived products. White chocolate has the same ingredient list as milk chocolate (w/o the cocoa solids), but again, the proportions and allowed/disallowed additives may be different. So you have to mind your brands/country-of-origin, as well as the "family" of chocolate.
Hershey's (US): http://www.thehersheycompany.com/nutrition-and-wellness/chocolate-101/types-of-chocolate.aspx
EU standards: http://europa.eu/legislation_summaries/consumers/product_labelling_and_packaging/l21122b_en.htm (If someone can find a more consumer-friendly explanation of EU chocolate labeling, that would be great.)
Chocolate and cocoa powder are two different things
For drinks go with cocoa powder (Dutch process). Emulsifying the fat in chocolate is pointless and not particularly tasty. Chocolate is about 40% to 60% fat, cocoa powder is 10% to 20% fat
For the chocolate milk taste you generally want the milk fat favours, not the cocoa fat
If your cocoa powder is not bright enough, adding a little coffee will help, as will a little more roasting of the cocoa powder, but be careful, it burns quickly. I think many commercial chocolate milks use a little salt (Sodium or Ammonium) as a brightener**
In some countries you can buy "extra" or "premium dutch process" cocoa powder, it is very dark in colour, and has a lot more of that dark chocolate style flavour
** The "that tastes nice and different" reaction
Best Answer
Spanish hot chocolate and Italian cioccolata fiorentina both use cornstarch as a thickening agent. Both are used more for dipping or sipping (churros in the former case), however you could easily just use less cornstarch to make it more 'drinkable'.
Try a teaspoon of cornstarch, mixed with a little cold water, added to the milk when you boil it.
As Kate Gregory suggests, you could also use full-fat milk, or loads of shaved chocolate.