Place the butter in a wide, light-colored pan over medium-low heat, and allow it to melt, then boil, then sizzle and turn nutty-brown.
Immediately pour the browned butter into a heat-safe dish, and allow it to cool and solidify.*
Toast the chopped pecans, either on a baking sheet in a 350 degree F oven, or in a dry skillet over medium heat, until browned and aromatic (about 5 to 10 minutes).
Preheat the oven to 350 degrees F and line baking sheets with parchment.
Place the solidified brown butter in a large mixing bowl, along with the brown sugar.
Beat on medium-high speed until very pale and fluffy (5 to 7 minutes minimum).
Add the egg, egg yolk, and vanilla, mixing until combined.
Scrape the bottom and sides of the bowl with a silicone spatula, then add the flour, toasted chopped pecans, cornstarch, baking soda, and salt, mixing just until combined (be careful not to overmix).
Use a 1.5-tablespoon scoop to portion the dough into balls, spacing them about 2-inches apart on the prepared baking sheets.
Sprinkle the unbaked cookies with demerara sugar and press a pecan half into each one (optional).
Bake for 8 to 10 minutes, or until set around the edges but still very soft towards the middle.
Cool on the baking sheets.