Make the Dough: In the bowl of a stand mixer with dough hook or paddle attachment, combine sourdough discard, warm milk, melted butter, egg, and vanilla. Add baking powder, baking soda, salt, and 2½ cups of flour. Mix until soft and slightly tacky, adding more flour 1 tablespoon at a time if needed. Let rest 10 minutes.
Make the Apple Filling: In a small saucepan, melt butter over medium heat. Add diced apples, brown sugar, cinnamon, nutmeg, lemon juice, salt, and flour. Stir until apples soften and mixture thickens. Remove from heat, stir in vanilla, and let cool completely.
Roll and Fill: On a lightly floured surface, roll dough into a rectangle about 10x28 inches using a rolling pin. Spread apple filling in an even layer over the dough.
Slice and Arrange: Roll dough tightly from the long side. Cut into 24–26 thin slices using unflavored dental floss or a sharp knife. Place into a prepared baking dish lined with parchment or lightly greased with butter.
Bake: Bake at 350°F for 22–24 minutes, until tops are golden brown and the bottom of the pan shows bubbling filling.
Make the Frosting: Brown butter in a small pan. Let cool slightly. In a large bowl or with a mixer, beat browned butter, softened cream cheese, powdered sugar, and vanilla extract until smooth and fluffy.
Frost and Serve: Spread frosting over warm rolls. Serve as-is or with a scoop of vanilla ice cream for an amazing dessert.