Preheat the oven to 375 degrees F and line baking sheets with parchment paper.
Melt the butter in a large, light-colored skillet over medium-low heat until bubbling.
After the bubbling subsides, gently cook the butter until toasty brown and fragrant, swirling the pan occasionally to promote even heating.
Pour the brown butter into a large, heat-safe mixing bowl, and stir in the brown sugar and granulated sugar.
Stir in the egg until fully incorporated.
Stir in the egg yolk and vanilla until smooth.
Stir in the flour, cornstarch, baking soda, and salt.
Fold in the chocolate chips and about ⅔ of the toffee chips.
Use a trigger scoop to measure 3 tablespoons of dough at a time, placing the dough balls on the prepared baking sheets with about an inch or two in between for spreading. Bake until set around the edges but still very moist in the centers (approx. 6 to 10 minutes).
Use a large, round cookie cutter or small bowl to coax the cookies into a more circular shape while still warm.
Garnish with the remaining toffee chips and sprinkle with sea salt (if desired).
Cool completely on the baking sheets.