Melt the butter in a medium saucepan over medium heat, stirring frequently, until deep golden brown with a nutty aroma (5 to 7 minutes). Reserve 4 tbsp in a bowl for the frosting; cool the remaining butter 20 minutes.
Cream the cooled browned butter with the granulated and brown sugars until light and fluffy (2 to 3 minutes).
Add the whole egg and mix 30 seconds; scrape the bowl. Add the egg yolk and repeat.
Mix in the honey, vanilla, and butter extract on low until combined.
Add the flour, salt, cornstarch, baking soda, baking powder, and cornmeal; mix until just combined.
Scoop the dough with a 4-tbsp cookie scoop onto a plate, cover with plastic wrap, and chill 30 minutes.
Preheat the oven to 350 F; line two baking sheets with parchment paper.
Space the cookie balls 2 inches apart on the sheets (5 per pan).
Bake 13 to 15 minutes until the edges are golden and the centers set.
Cool on the sheets 5 minutes, then transfer to a wire rack.
For the honey buttercream: beat the cooled browned butter with the 2 tbsp softened butter until smooth. Add the powdered sugar on low until incorporated, then mix in the honey and salt on medium until fluffy (freeze 10 minutes if runny).
Once the cookies are completely cool, dollop 1 tbsp buttercream on each cookie's center and drizzle with honey before serving.
