Preheat oven to 375 F. Line baking sheets with parchment paper or silicone baking mat.
Melt butter in a heavy saucepan. The butter should be just melted. If you want to brown the butter, please see the note below.
Add brown sugar to a large bowl.
Pour melted butter over sugar.
Stir to combine butter and sugar. Let stand 5 minutes.
Add egg and egg yolk; stir to combine.
Stir in vanilla extract.
Add flour, baking soda, and salt. Stir until just combined. Please see the comments about flour in the recipe notes.
Stir in chocolate chips.
Using a medium cookie scoop (1.5 tablespoon), drop scoops of dough 2 inches apart on prepared cookie sheet.
Bake for about 7-10 minutes, or until the edges of the cookies are golden brown. The centers should still look soft and slightly underbaked. That will help to keep the cookies soft.
Let cool on cookie sheet for 5 minutes. Remove to wire rack to finish cooling.
Butter: If you want to brown the butter, you will need to use more butter because browning butter causes it to cook down. Otherwise, you will be left with a drier dough. If browning, you will need to use approximately 3 tablespoons more butter (so about 19 tablespoons or 269 grams total).
Correctly measuring the ingredients is key! I suggest either weighing the ingredients or aerating the flour (sifting or fluffing) and then scooping and leveling. If you don't want to sift or weigh the flour, please use less flour. Packing in too much flour will cause the cookies not to spread.
Can substitute 4 ounces finely chopped semisweet baking chocolate for chocolate chips.
I suggest stirring this recipe together by hand rather than using a hand or stand mixer.