Melt ¾ cup butter in a saucepan over medium-low heat.
Cook the butter, swirling the pan occasionally, until it becomes foamy and develops a deep amber color and nutty aroma (about 5 minutes).
Transfer the butter to a large heat-proof mixing bowl.
Add the chopped pecans and stir to coat. Cool to room temperature.
In a separate bowl, combine the flour, cornstarch, baking soda, and salt. Set aside.
Use a slotted spoon to remove the chopped pecans from the brown butter and place them in a separate bowl.
Add the brown sugar to the brown butter and use a hand or stand mixer to combine on medium-low speed.
Add the egg and vanilla and increase the mixer speed to high, beating until the mixture is smooth and light in color.
Using medium-low speed, mix in the dry ingredients into the brown sugar/butter mixture, followed by the chopped pecans, until the dough comes together.
Use a medium cookie scoop or your hands to form 24 balls from the dough, placing them on a plate once rolled.
Cover the plate with plastic and refrigerate the dough balls for at least 1 hour.
While the dough is chilling, heat the remaining tablespoon of butter and the maple syrup in a small saute pan over medium heat.
Once the butter and maple syrup are foamy, add the pecan halves and stir to coat for 1 minute. Remove from heat and cool.
Preheat the oven to 350 F.
Line 2 baking sheets with parchment paper.
Place the chilled dough on the baking sheets in 3 rows of 4 balls and use a press or bottom of a glass to slightly flatten each ball.
Top each cookie with one of the glazed pecans.
Bake the cookies, 1 sheet at a time, on the middle rack of the oven for 9 to 10 minutes, until the edges are golden brown but the centers are still slightly soft. Be careful not to overbake.
Allow the cookies to cool on the baking sheet for 5 minutes, then transfer to a cooling rack.
Serve the cookies immediately or store in an airtight container at room temperature.