Applesauce Spice Cake

Place oven rack in middle position and preheat oven to 350°F. Butter pan and set aside.

Whisk together flour, baking soda, spices, and salt in a bowl.

Beat together sugar, butter, and rum with an electric mixer at medium-high speed until combined well, then add egg and beat until pale and fluffy, 2 to 3 minutes with a stand mixer or 5 to 6 minutes with a handheld. Reduce speed to low and add dry ingredients, mixing until combined well.

Add applesauce and mix until combined well.

Spread batter evenly in springform pan and bake until a wooden pick or skewer comes out clean, 30 to 35 minutes.

Cool cake in pan on a rack 10 minutes, then remove side of pan and cool completely.

Melt 2 tablespoons butter in a 1 1/2- to 2-quart heavy saucepan, then add sugar and evaporated milk and simmer, stirring constantly until sugar is dissolved, about 4 minutes.

Remove from heat and whisk in rum, vanilla, salt, and remaining tablespoon butter, then whisk in confectioners sugar 1 tablespoon at a time. Cool to warm, about 20 minutes, then spread over cooled cake.