To prepare cake, melt 2 tablespoons butter in a large skillet over medium-high heat. Add apples to pan; sauté until just softened, about 3 minutes. Stir in brown sugar and ⅛ teaspoon salt; sauté until apples release liquid and most of liquid evaporates, about 6 minutes. Remove from heat and stir in 1 ½ teaspoons cinnamon. Cool slightly.

Whisk together flour, baking powder, baking soda, allspice, and remaining 1 teaspoon salt and ¼ teaspoon cinnamon in a medium bowl.

Place softened butter, cream cheese, and granulated sugar in a large bowl; beat with an electric mixer until creamy, about 2 minutes. Beat in vanilla. With mixer on medium-low speed, add eggs, 1 at a time, beating well after each addition. Add flour mixture and buttermilk alternately to sugar mixture, beginning and ending with flour mixture. Beat until batter is smooth.

Fold half of apple mixture into batter.

Pour batter into prepared pan. Spoon remaining apple mixture evenly over batter.

Bake at 350°F until a wooden pick inserted in center comes out with a few moist crumbs, 40 to 43 minutes.

Cool cake in pan for 20 minutes. Carefully remove cake from pan, using parchment to lift it out. Cool cake completely on a wire rack, about 1 hour.

To prepare glaze, whisk together powdered sugar, milk, vanilla, and salt in a medium bowl until mixture is about the consistency of whipping cream.

Drizzle glaze evenly over cake.