Preheat the oven to 375°F. Line two large baking sheets with parchment paper. Place the turbinado sugar in a shallow bowl.
In the bowl of a stand mixer fitted with the paddle attachment, or using a hand mixer, beat together the softened butter, peanut butter, granulated sugar, and brown sugar on medium speed for 2–3 minutes, until light and fluffy. Scrape down the sides of the bowl.
Add the eggs one at a time, mixing well after each addition. Mix in the vanilla extract until fully incorporated.
Add the flour, cornstarch, baking soda, and kosher salt. Mix on low speed just until no dry streaks remain. Do not overmix.
Using a large cookie scoop, about 3 tablespoons per cookie, portion the dough. Roll each dough ball in the turbinado sugar until evenly coated.
Place the cookies on the prepared baking sheets, leaving about 2½ inches between each cookie. There is no need to flatten the dough.
Bake for 10–11 minutes, or until the edges are lightly golden and the centers still appear slightly underbaked.
Immediately sprinkle the warm cookies with flaky sea salt.
Allow the cookies to rest on the baking sheet for 10 minutes before transferring them to a wire rack to cool completely.