Grease an 8x8x2-inch baking pan. Line bottom with parchment or waxed paper. Grease paper; set pan aside. Combine flour, granulated sugar, baking powder, baking soda, and salt in a large bowl.
Combine egg, carrots, and oil in another mixing bowl. Add egg mixture to flour mixture, stirring until combined. Spread batter into prepared pan.
Bake in a 350 degree F oven for 30 to 35 minutes or until a toothpick inserted near the center comes out clean. Cool on a wire rack for 10 minutes. Loosen edges and remove cake from pan; remove paper. Turn right side up. Cool thoroughly.
Heat butter in a medium saucepan until melted. Remove from heat; stir in sifted powdered sugar and brandy, orange liqueur, or vanilla. Stir in enough water, 1 teaspoon at a time, to make of glazing consistency.
Cut cake into nine squares. Cut each square in half diagonally to form triangles. Place cake pieces on a rack over a baking sheet. Slowly spoon brandy glaze over each piece, allowing some glaze to drizzle down sides. Top with chopped toasted pecans or hazelnuts, if desired. Makes 18 servings.