Melt the butter in a light-colored skillet over medium heat, swirling often, for about 4 minutes until foamy and just starting to turn golden. Add the pecans and cook, stirring constantly, for 2-3 minutes until the butter is deep golden with nutty brown bits and the pecans are toasted.Pour everything into a large bowl and cool for 15 minutes.
Whisk the brown sugar and granulated sugar into the pecan butter until combined. Whisk in the milk and vanilla for about 30 seconds, until glossy.
In a separate bowl, whisk together the flour, cornstarch, baking soda and salt. Stir into the butter mixture just until combined, then fold in the toffee bits.
Cover and chill the dough for 45 minutes. This firms up the brown butter so the cookies bake thick and chewy.
Preheat the oven to 350°F and line two baking sheets with parchment paper. Scoop the dough into 2 tbsp mounds and place them 2 inches apart.
Bake for 10-12 minutes, until the edges are golden and the centers are soft and puffy. If the toffee spreads at the edges, gently nudge each hot cookie back into a circle with a large round cookie cutter or the rim of a glass.
Cool on the baking sheet for 10 minutes, then transfer to a wire rack. Makes about 16 cookies.