Use a handheld electric mixer to beat together the cream together the butter, cream cheese, and sugar in a large bowl. Beat in the eggs and vanilla. Add the flour, cornstarch, baking powder, baking soda, and salt, and mix until well-combined. The dough will look wet.
Lay a piece of plastic wrap directly on top of the cookie dough and refrigerate until the dough is stiffened, about 2 hours.
Preheat the oven to 350F. Line 2 half sheet pans with parchment paper or silpat liners.
Use a 1 1/2-tablespoon scoop to measure out the dough. Roll it into balls and arrange them 2 inches apart on the baking pans (8 cookies per half sheet pan works well).
Use a glass to slightly flatten each cookie dough ball.
Bake until the cookies are light golden on the bottom, puffed on top, and just stop looking doughy in the center, rotating the trays once halfway through. This takes about 9 to 12 minutes. Be careful not to overbake, you don’t want these cookies golden on the top. Repeat with the remaining dough.
Let the cookies cool for 5 minutes on the trays, and then use a thin metal spatula to transfer them to a wire rack to finish cooling completely.