Preheat oven to 350°F. Line a 12-cup muffin pan with paper or foil cupcake liners.
Peel, quarter and core apples and cut into 1/3-inch dice. Melt butter in a medium skillet over medium heat. Add apples; cook for about 5 minutes. Remove from heat; stir in brown sugar and salt. Let cool for 5 minutes.
In a bowl, whisk together flour, walnuts and baking soda. In another bowl, beat together egg and vanilla extract. Stir apple and egg mixtures into flour. Spoon batter into cupcake liners about three-quarters full.
Bake for 18 to 20 minutes. Let cool for 15 minutes. Remove from pan to racks to cool completely. Dust with confectioners' sugar before serving.