In a light-colored saucepan over medium heat, melt the butter. Continue cooking, swirling the pan, until it foams and golden-brown solids form at the bottom and it smells nutty. Immediately pour into a large, heatproof bowl.

Add the brown sugar and granulated sugar to the hot brown butter and whisk until combined. Whisk in the egg, egg yolk, and vanilla extract until smooth.

In a separate bowl, whisk together the flour, cornstarch, baking soda, and salt. Add the dry ingredients to the wet ingredients and mix until just combined.

Gently fold in the chocolate chunks and toffee bits.

Cover the bowl and chill the dough in the refrigerator for at least 2 hours, or up to 48 hours.

Preheat the oven to 375°F (190°C). Line two baking sheets with parchment paper.

Scoop dough into 2-tablespoon balls and place them 2-3 inches apart on the baking sheets. Bake for 10-12 minutes, until the edges are golden brown but the centers are still soft. Let them cool on the baking sheet for 5 minutes before transferring to a wire rack.