How can I get my chocolate chip cookies to come out thick and soft? I always use real butter; is this a problem?
I've found the key to be chilling the dough. If you can get it really cold, put it on room temperature baking sheets, and put it in a well preheated oven, you get a lot less spread and therefore a considerably thicker, cakier cookie, which I also prefer. If I mix the batch in my KitchenAid, I'll put the metal bowl in the coldest part of the refrigerator for 30 minutes to an hour before portioning and baking, with great results.
In the Good Eats episode, "Three Chips for Sister Martha", he referred to what I think you're asking for as the 'puffy' cookie. See the transcript for how he varied a basic recipe to get more puffy / chewy / crispy variations.
For the puffy ones, he recommended shortening, as it'd melt later, giving more rise. So yes, the butter might be a problem. There is on the market butter flavored shortening, if you want the taste. (and it comes in sticks, making it easy to measure ... I use it for my christmas cookie baking, as I've got a few vegan friends).
He also recommended more brown sugar for tenderness, baking powder instead of baking soda to reduce the amount it spreads, cake flour instead of all purpose flour, and chilling the batter so it spread less. See the transcript for more details.
For a cookie with a texture more like cake, make sure to use soft room temperature butter, but don't melt it. It should be around 70 degrees. Cream the butter with white sugar. Make sure to use whole eggs - even extra whites, as foamy whites will add to the bubble network needed for a cake-y texture.
This is the opposite of the Cook's Illustrated cookie, which is designed to be chewier.
In Cook's Illustrated Chewy Chocolate Cookies (different recipe than the one above), the tricks include only using an egg white, and replacing some of the sugar with dark brown corn syrup because the yolk surprisingly dries it out. They also suggest using softened butter that's around 70 degrees, and chilling for thirty minutes before doling out.
There's so much science in baking cookies. Yolk vs. white vs. whole egg is only one part of the equation. Changing the egg won't necessarily make or break the spread of the cookie.