Sift together the cake flour, bread flour, baking soda, baking powder and salt into a large bowl and set aside.
Cream together the butter and sugars on medium speed until very light, about 5 minutes. Add the egg and the egg yolk one at a time, mixing well after each addition, then add the vanilla. Reduce the mixer speed to low, gradually add the dry ingredients and mix until just combined, 5 to 10 seconds. Using a rubber spatula, fold in the chocolate chips.
Press plastic wrap against the dough and refrigerate for at least 2 hours, up to 24 hours.
When ready to bake, preheat oven to 350 degrees F. Line a baking sheet with parchment paper or a nonstick baking mat.
Scoop 2 ounces of dough, roll into a rough ball (it should be a little larger than a golf ball) and place on the baking sheet, leaving at least 2 inches of space between balls of dough. Bake until the edges of the cookies are light brown and set, but the centers still look pale, about 13 to 16 minutes. Cool the cookies on the baking sheet for 10 minutes, then transfer to a wire rack to cool completely. Leftover cookies can be stored in an airtight container at room temperature for up to 3 days, or wrapped in plastic wrap, placed in a freezer bag and frozen for up to 3 months.
You can substitute all-purpose flour for the bread flour and cake flour; if you do so, use 3⅓ cups of all-purpose flour.