In a large heavy saucepan, bring brown sugar, butter, cream, syrup, and salt to a boil over medium heat, stirring constantly. Reduce heat to medium-low; cook 5 minutes, stirring occasionally. Remove from heat.
Gradually add confectioners’ sugar, vinegar, and vanilla, beating at medium speed with an electric mixer until smooth. Let stand until mixture begins to thicken and is slightly cooled, approximately 30 minutes, stirring occasionally. Finely chop 1 cup Buttered Pecans; set aside. Reserve remaining 2 cups pecan halves.
Buttered Pecans
Preheat oven to 350°. On a rimmed baking sheet, place pecans, butter, and salt. Bake until toasted, approximately 12 minutes, stirring occasionally. Remove from oven. Let cool completely.
Cake assembly and frosting
On a cake plate or stand, place one cake layer; top with approximately 3⁄4 cup frosting. Gently spread with an offset spatula, leaving a 1⁄2-inch border, until frosting thickens, 1 to 2 minutes. Sprinkle evenly with half of finely chopped Buttered Pecans. Repeat procedure once. Top with remaining cake layer; spread with approximately 3⁄4 cup frosting. Secure center of cake with a wooden pick. Spread a thin layer of frosting around sides of cake; let stand until set, approximately 20 minutes. Frost top and sides of cake with desired amount of remaining frosting. Remove wooden pick just before serving. Top cake with reserved Buttered Pecans. Sprinkle with 1⁄4 teaspoon kosher salt, if desired.