Beat melted butter and both sugars with an electric mixer for 1 minute. Add egg and beat again for 30 seconds. Let rest for 5 minutes, then beat again for another minute. Beat in the vanilla.
Mix together the flour, baking soda and salt, then add the flour mixture and stir to form a soft dough. Stir in the chocolate. Dough should be fairly soft at this point.
Chill dough for about 10 minutes or just until it is firm enough to scoop. Scoop about ¼ cup batter for each cookie onto a couple of foil lined plates or a baking tray (8-10 total). Cover loosely with plastic wrap and chill for two hours or until dough blobs are firm enough so that you can shape them into neater rounds. When they are firm, shape them into neater balls and press down slightly so you have discs. For best results chill overnight, or go ahead and bake.
Preheat oven to 375°F. Line a large baking sheet with parchment paper. Arrange cookies 3 inches apart on the sheet. Put in the oven.
Put the cookies in the 375 degree oven, then close door and reduce heat to 325 degrees F.
Bake cookies for about 15 to 18 minutes, checking at 10 and rotating the pan.
If using milk chocolate, pull from oven and immediately stick milk chocolate pieces into the hot cookies.
Let cool on baking sheet for about 5 minutes, then transfer to a wire rack and let cool completely.
Notes
If you use any flour other than cake flour, make sure to weigh it.