A few years ago when I had my gluten intolerance, I modified the Katharine Hepburn brownie recipe not to use any wheat flour. I chose that particular recipe because A) it's the best brownie recipe I've ever tried, and B) there's so little flour in it that I figured the modification had a chance at being pretty successful.
I tried rice flour (too gritty) and cornstarch (too off), but settled on potato starch (just right, and very moist). Which makes it a perfect recipe for Passover, too.
These are very dense rich, moist brownies. I wouldn't make them with wheat flour anymore, because the potato starch keeps the moistness much better, IMO.
KFP/GF Brownies You'll Make All Year And For People Who Can Eat Wheat, Too
1 stick butter
1/2 cup cocoa powder (the better the cocoa powder, the better the brownies)
1 cup sugar
1/4 potato starch
pinch of salt
splash of pure vanilla extract
Preheat the over to 325 degrees F. Grease an 8-inch by 8-inch baking pan (or line the bottom with parchment paper).
In a small saucepan, melt together the butter and cocoa powder. Stir until smooth. Remove from heat and allow to cool for a few minutes.
Stir the eggs in quickly, one at a time, so they don't scramble. The mix will look really gloppy at this point. Stir in the sugar, and it should smooth out considerably. Stir in the potato starch and salt, and then the vanilla extract.
Pour into the baking pan, and bake for 30 minutes. This should be just about underbaked (better to underbake than overbake), but check at 25 minutes.
Allow to cool for as long as you can stand it, then eat.