In a medium saucepan, melt the butter over medium-low heat. As the butter melts, it’ll start to foam, then turn a clear golden color. Continue to stir the butter, scraping up any browned specks from the bottom of the pan as it starts to brown. It should have a nutty aroma. Once browned and nutty, remove from the heat, and immediately pour into a shallow bowl to halt cooking.
Whisk the flour, baking soda, salt, and cornstarch together in a medium bowl; set aside.
In a stand mixer fitted with the paddle attachment, beat the browned butter and both sugars together on medium speed until smooth and incorporated, about 1 minute. Add the eggs and vanilla and continue beating until combined, using a rubber spatula to scrape down the sides of the bowl as needed. Add the dry ingredients and mix on low until just incorporated.
Take the toffee out of the freezer and chop into small pieces. Gently fold most of the toffee into the dough, reserving some for the tops of the cookies after they've baked. Cover and refrigerate dough for at least two hours, or up to three days.
Preheat the oven to 350ºF and line a baking sheet with parchment paper. Roll the dough into balls about the size of a golf ball, then place them on the baking sheet about 2 inches apart. Flatten slightly to create a thick "puck".
Bake for 8-10 minutes, or until the edges are set and the centers are puffy but still look slightly underdone. Immediately after removing the cookies from the oven, press reserved toffee pieces into the tops of the cookies. Allow cookies to cool on the baking sheet for 10 minutes before transferring to a wire rack to cool completely.