Stir apples, brown sugar, butter, and salt in a large skillet over medium-high heat. Cook, stirring occasionally, until soft and syrupy, about 30 minutes. Stir in walnuts.
Preheat the oven to 375 degrees F (190 degrees C). Grease a muffin tin; divide apple mixture among muffin cups.
Whisk flour, brown sugar, baking powder, apple pie spice, baking soda, and salt together in a large bowl. Make a small well in the center and set aside.
Combine butter, applesauce, buttermilk, egg, and vanilla extract. Pour into the well in the flour mixture. Mix until just combined; batter should be lumpy. Pour batter over apple mixture in the muffin cups.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 15 to 20 minutes. Cool on a wire rack for 5 minutes. Run a knife around the edges of each muffin. Carefully flip pan onto a serving platter; serve hot.