Grease (with shortening) and flour 3 (9-inch) round cake pans; line with parchment paper, and grease parchment.

Preheat oven to 350°F. Cook 1 cup of the pecans and 1 1/2 tablespoons of the butter in a small skillet over medium, stirring, until toasted, 6 to 8 minutes. Spread pecans on wax paper; cool about 10 minutes.

Meanwhile, beat remaining 1 cup butter with a heavy-duty stand mixer on medium speed until creamy; gradually add sugar, beating until light and fluffy. Add egg yolks, 1 at a time, beating just until blended after each addition. Stir in vanilla.

Stir together flour, baking soda, and salt; add to butter mixture alternately with buttermilk, beginning and ending with flour mixture. Beat at low speed until blended after each addition. Stir in cooled butter-toasted pecans.

Beat egg whites on medium speed until stiff peaks form; fold one-third of egg whites into batter. Gently fold in remaining beaten egg whites just until blended. Pour into prepared pans.

Bake in preheated oven until a wooden pick inserted in center comes out clean, 20 to 22 minutes. Cool in pans on wire racks 10 minutes. Remove from pans, and cool completely on wire racks, about 20 minutes.

Meanwhile, spread remaining 1/2 cup finely chopped pecans on a baking sheet and bake at 350°F until toasted, about 5 minutes.

Prepare the Browned Butter Frosting: Cook 1 1/2 cups salted butter in a medium-size heavy saucepan over medium, stirring occasionally, until butter is fragrant and milk solids turn golden brown, 10 to 12 minutes. Remove pan from heat, and immediately pour butter into a small bowl. Cover and chill until butter is cool and almost firm, about 1 hour, stirring every 15 minutes. (If butter becomes solid, it will soften again once you begin beating it.) Beat browned butter with a heavy-duty electric stand mixer fitted with the paddle attachment on medium speed until smooth and fluffy, about 3 minutes. Gradually add 6 cups powdered sugar alternately with 1/3 cup whole milk, beginning and ending with sugar. Beat on low speed until well blended after each addition. Beat in 1 1/2 tsp. vanilla extract.

Spread Browned Butter Frosting between layers and on top and sides of cake. Gently press toasted pecans around bottom of cake about 1 inch up the sides.