Cut figs in half; cover with hot water. Let stand 20 minutes; drain and set aside.
Beat 1 cup butter at medium speed with an electric mixer about 2 minutes or until creamy; gradually add sugars, beating well. Add eggs, 1 at a time, beating until blended after each addition.
Combine flour, baking soda, and salt; add to butter mixture alternately with sour cream, beginning and ending with flour mixture. Beat at low speed until blended after each addition. Stir in 1 teaspoon vanilla. Pour batter into a greased and floured 10-inch springform pan. Top with fig halves.
Bake at 300° for 1 hour and 50 minutes or until a wooden pick inserted in center comes out clean. Cool in pan on wire rack 10 minutes; remove from pan, and cool on rack.
Melt 2 tablespoons butter in a small saucepan over medium heat; add sugar and cinnamon, stirring until sugar dissolves. Stir in 1/4 teaspoon vanilla. Brush over top of cake. Garnish, if desired.