To make the filling
Put the butter in a small saucepan over medium to medium-high heat. Allow the butter to melt, then continue to cook, stirring constantly, until the butter solids are browned and smell toasty, about 3 minutes or more. Watch carefully so the butter does not burn. As the butter browns, the foam rises to the top and dark brown particles stick to the bottom of the pan. As soon as the butter is a dark golden brown, remove it from the heat and pour it into a small bowl.
In the bowl of a food processor, combine the brown butter, pecans, powdered sugar, vanilla extract, and salt and pulse until a thick paste forms, scraping down the bowl of the food processor a few times during the pulsing. Remove the filling from the food proccessor, cover, and chill while you are making the cookie dough.
To make the cookies
To make the cookie dough, sift the flour, cocoa, and baking soda together into a bowl. Set the dry ingredients aside.
In the bowl of an electric mixer with the paddle attachment, combine the butter and sugar and cream together on medium-high speed until very pale and fluffy, about 3 minutes. Scrape down the bowl.
Add the egg on medium-low speed and beat to incorporate.
Scrape the bowl and the paddle and add the pecans, vanilla extract, and salt.
Add the dry ingredients on low speed in 3 or 4 additions and mix until just combined. Do not over mix. Remove the bowl from the mixer and scrape down all the way to the bottom of the bowl with a rubber spatula to be sure everything is well mixed.
Chill the cookie dough for 1 hour or longer before shaping.
When you are ready to shape and bake the cookies, preheat the oven to 375°F. Portion the dough using a 1⁄3 oz. ice cream scoop. (If you don’t have a scoop, use about 2 tsp. of dough per cookie.) You should get 48 cookies. Place the cookies 3⁄4 to 1 inch apart on parchment-lined baking sheets and use your fingers to press them flat, a little less than 1⁄4 inch thick. (You should be able to get 16 cookies on each baking sheet and bake in 3 batches.) Bake the cookies, one sheet pan at a time, until the cookies are cooked through and feel set, about 7 minutes. Do not open the oven door while they are baking. Remove the pan from the oven and allow to cool for at least 10 minutes before removing the cookies with a metal spatula. Cool the cookies completely before filling.
To make a cookie sandwich, turn one cookie flat side up and spread with about 2 tsp. of filling (you can use the same small ice cream scoop that you used for the cookie dough to portion out the filling), then top with another cookie, flat side down, pressing gently. Repeat until all the cookies are sandwiched together.