3 cups peeled and finely chopped Gala apples (about ½ pound)
½ cup firmly packed light brown sugar
¼ cup butter
3 Tbsp milk
1 tsp vanilla extract
1 cup powdered sugar
Prepare filling: beat first 3 ingredients at medium speed with an electric mixer until blended and smooth. Add egg, flour, and vanilla; beat just until blended.
Prepare batter: preheat oven to 350 degrees. Bake pecans in a shallow pan 8 to 10 minutes or until toasted and fragrant, stirring halfway through. Stir together 3 cups flour and next 7 ingredients in a large bowl; stir in eggs and next 3 ingredients, stirring just until dry ingredients are moistened. Stir in apples and pecans. Spoon two-thirds of apple mixture into a greased and floured 14 cup Bundt pan. Spoon cream cheese filling over apple mixture, leaving a 1-inch border around edges of pan. Swirl filling through apple mixture using a paring knife. Spoon remaining apple mixture over cream cheese filling. Bake at 350 degrees for 1 hour or until a long wooden pick inserted in center comes out clean. Cool cake in pan on wire rack 15 minutes; remove from pan to wire rack; and cool completely (about 2 hours).
Prepare frosting: bring ½ cup brown sugar, ¼ cup butter, and 3 Tbsp milk to a boil in a 2-qt sauce pan over medium heat, whisking constantly, boil 1 minute, whisking constantly. Remove from heat; stir in vanilla. Gradually whisk in powdered sugar until smooth; stir gently 3 to 5 minutes or until mixture begins to cool and thickens slightly. Pour immediately over cooled cake.