Preheat oven to 350 degrees. Spray 2 (9-inch) cake pans with cooking spray with flour. Line bottom of pans with parchment paper and lightly spray parchment paper.
Sprinkle the chopped pecans in the bottoms of the pans.
Using an electric mixer, beat butter and dark brown sugar at medium speed until fluffy, about 3 to 4 minutes. Be sure to scrape down the sides of the bowl to evenly mix everything.
Add eggs one at a time, beating well after each addition.
In a medium bowl, whisk together flour, baking powder, and salt.
Add flour mixture, alternating with milk, beating just until combined after each addition. (Add flour mixture in 3 additions and milk in 2).
Beat in vanilla.
Divide batter evenly between the prepared pans, smoothing out the top with a spatula.
Bake 32 to 36 minutes or until a wooden pick inserted in the center comes out clean. Let cool in pans 10 minutes and then remove from pans to cool completely.
Make Filling
In a medium saucepan, stir together pecans, sugar, corn syrup, melted butter, egg, egg yolk, vanilla, and salt.
Bring to a boil over medium-high heat, stirring frequently. Simmer and stir for about 3 minutes.
Pour into a bowl and let cool to room temperature.
Spread pecan filling on top of one cake layer. Place the second layer on top.
Make Brown Sugar Frosting
In a small saucepan, bring brown sugar, water, and salt to a boil. Stir until sugar is dissolved. Add 1/2 cup butter, stirring until melted. Let cool to room temperature.
Combine cooled brown sugar mixture and remaining butter in a mixing bowl and beat with an electric mixer until smooth.
Gradually beat in powdered sugar.
Beat in milk. Spread frosting on top and sides of cake.