Melt 1 teaspoon butter in bottom of a 9-inch round cake pan. Combine oats, 3 tablespoons brown sugar, and pecans. Sprinkle oat mixture over bottom of pan, and set aside.
Beat 3 tablespoons butter, granulated sugar, and 1/4 cup brown sugar at medium speed of a mixer until mixture is well-blended (about 4 minutes). Add the egg, and beat well. Lightly spoon the flour into dry measuring cups, and level with a knife. Combine flour, baking powder, pumpkin-pie spice, and baking soda. Add the flour mixture to the butter mixture alternately with the pumpkin and buttermilk, beginning and ending with the flour mixture.
Spoon batter evenly over oat mixture. Bake at 350° for 40 minutes or until a wooden pick inserted in center comes out clean. Cool in pan 5 minutes, and invert cake onto a serving plate. Serve warm or at room temperature.