Preheat the oven to 350 degrees F. Line a 12-cup muffin pan with nonstick liners. Peel and core the apple and cut into 12 wedges, then slice the wedges crosswise into small triangular pieces. Combine the apple pieces, 1 teaspoon granulated sugar, 1 tablespoon melted butter and a pinch each of cinnamon and salt in a large nonstick skillet. Cook over medium-high heat, stirring occasionally, until the apples are browned and softened, 4 to 6 minutes. Let cool completely.
Whisk the flour, baking powder, remaining 1 teaspoon cinnamon and 1/2 teaspoon salt in a large bowl. Whisk the remaining 1/2 cup granulated sugar and 6 tablespoons melted butter, 1 cup milk, the pumpkin, 1 teaspoon vanilla and the eggs in a medium bowl; stir into the flour mixture until just combined.
Divide the batter among the muffin cups, filling them three-quarters of the way. Spoon the apples into the center of each. Bake until a toothpick inserted into the center of a muffin comes out clean, 22 to 25 minutes. Let the muffins cool 5 minutes in the pan, then remove to a rack to cool completely.
Meanwhile, make the glaze: Stir the confectioners' sugar with 1 1/2 teaspoons milk and the remaining 1/4 teaspoon vanilla in a small bowl until smooth, gradually adding up to 1 more teaspoon milk if needed. Drizzle the glaze over the muffins. Let set 10 minutes.
Photograph by Ryan Dausch