Place a non-stick skillet over medium-high heat and heat the pan dry. Reduce the heat to medium after the pan is GOOD AND HOT and add the chopped pecans. Shake the skillet over the heat for 1 to 2 minutes or until you begin to smell the pecans. Remove the skillet from the heat and immediately add in 1 1/2 tablespoons of unsalted butter and continue shaking the skillet as the butter melts to coat the nuts.
Return the skillet to medium heat and sauté the chopped pecans and butter until fragrant and toasted, about 3 to 5 minutes. Watch the smaller bits of nuts as they will brown/cook much faster and could burn if not continuously stirred. Remove from the heat and allow to cool.
In a large bowl, whisk together the all-purpose flour, cornstarch, baking soda, and salt. Set aside.
Brown the remaining 1 cup of butter. Heat the butter over medium-high heat until the melting process is well underway. Reduce the heat to medium and cook until the butter first foams, then until the milk solids become a golden-brown color, about 15 to 20 minutes depending on your stove.
Once the butter is sufficiently brown, no longer foaming, and is fragrant with no visible milk solids remaining, pour the butter into a large mixing bowl and cool 10 minutes. Whisk both granulated and brown sugars into the cooled butter. Then, whisk in two large room temperature eggs and some vanilla until well combined.
Combine the butter mixture and flour mixture either from one bowl to another or in the bowl of a stand mixer with the paddle attachment. Stir until no streaks of flour remain, then stir in the toasted chopped pecans and all of their butter. Cover the surface of the dough with plastic film to prevent any air in between. Chill for 2 hours or up to 3 days.
One hour before assembling to bake, remove the dough from refrigeration. Arrange your oven racks to the middle position. Preheat your oven to 350°F. Have cookie sheets prepared by lining with parchment paper or Silpats.
Using a large cookie scoop, about a 3-tablespoon scoop, manage 18 cookies that are as similar in size to one another as possible. Place 9 cookies spaced 2 inches apart on the prepared baking sheets.
Dust the tops of the cookies with a pinch of granulated sugar and gently press a single pecan half into each center. Bake for 9 to 12 minutes or until the edges are set and the cookies are just beginning to turn a golden-brown.
Remove the baking sheets from the oven and let the cookies begin to cool directly on the baking sheets for 5 minutes. If you are using flakey sea salt, add it while the cookies are still hot and resting on the baking sheets.
Use a metal spatula with a thin blade to carefully lift the cookies from the baking sheets to a wire rack to finish cooling, about 30 to 45 minutes. The cookies must be completely cooled before they may be loaded into cookie tins or placed alongside other cookies on a cookie tray. If in doubt, just give it an extra minute or two.