Prepare the Pecan Pie Filling: Whisk together corn syrup, brown sugar, cornstarch, egg yolks, half-and-half, and salt in a heavy 3-quart saucepan until smooth. Bring mixture to a boil over medium, whisking constantly; boil, whisking constantly, until thickened, about 1 minute. Remove from heat; whisk in butter and vanilla, and fold in pecans. Place wax paper or plastic wrap directly on surface of mixture to prevent a film from forming, and chill 4 hours.

Prepare the Pecan Pie Cake: Preheat oven to 350°F. Grease (with cooking spray) and flour 3 (8-inch) round cake pans and line bottoms with parchment paper; lightly grease parchment paper with cooking spray. Beat butter with a stand mixer fitted with paddle attachment on medium speed until creamy, 1 to 2 minutes. Gradually add granulated sugar and brown sugar, beating until light and fluffy, 3 to 4 minutes. Add egg yolks, 1 at a time, beating just until blended after each addition. Beat in vanilla just until blended.

Whisk together flour, baking soda, and salt in a medium bowl; add to butter mixture alternately with buttermilk, beginning and ending with flour mixture. Beat on low speed just until blended after each addition. Gently fold in pecans.

Beat egg whites with an electric mixer 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 batter evenly into prepared pans.

Bake in preheated oven until a wooden pick inserted in center comes out clean, 22 to 25 minutes. Cool in pans on wire racks 10 minutes; remove from pans, and discard parchment paper. Cool completely on wire racks, about 1 hour.

Meanwhile, prepare the Browned Butter Frosting: Cook butter in a small heavy saucepan over medium, stirring constantly, until butter melts and begins to turn golden brown, 6 to 8 minutes. Remove pan from heat immediately, and pour butter into a small heatproof glass bowl. Cool 10 minutes at room temperature. Cover and chill until butter begins to solidify, about 1 hour.

Beat chilled browned butter with stand mixer fitted with paddle attachment on medium speed until fluffy, about 1 to 2 minutes. Gradually add powdered sugar alternately with 5 tablespoons cream, beginning and ending with powdered sugar. Beat mixture on low speed until well blended after each addition. Beat in vanilla. If needed, add up to 1 tablespoon cream, 1 teaspoon at a time, and beat until desired consistency is reached.

Assemble cake: Place 1 cake layer on a cake plate or platter. Spoon 1 cup buttercream into a pastry bag fitted with a star tip (2D), or spoon frosting into a heavy-duty ziplock plastic freezer bag, and snip 1 corner to make a small hole. Pipe a ring of frosting around cake layer just inside top edge. Spread cake layer with about 1 cup chilled filling, spreading to edge of piped frosting. Repeat with remaining 2 cake layers. Spread remaining frosting over sides of cake. Pipe any remaining frosting in a decorative pattern on top of cake. Chill at least 1 hour or up to 48 hours. Let stand at room temperature 30 minutes before serving. Store leftovers, covered, in refrigerator.