For the Spiced Apples: In a 10-inch skillet, melt butter over medium heat. Once melted, add apple pieces, brown sugar, cinnamon, salt, and pie spice and cook, stirring frequently, until apples are tender, about 10 minutes.

Transfer apple mixture to a fine mesh strainer set over a medium bowl. Thoroughly strain, gently stirring to remove as much excess moisture from apples as possible; set aside to let apples strain until they reach room temperature, about 15 minutes. Spread strained apples in an even layer on a large plate and refrigerate uncovered until cooled, 15 to 20 minutes. Reserve strained sauce for use as a glaze.

For the Muffins: Adjust oven rack to middle position and preheat oven to 400°F (205℃). Line a 12-cup muffin tin with paper liners. Stir together brown sugar, pie spice, ½ cup (2.25 ounces; 64g) of the flour, and 1 ½ teaspoons of the cinnamon in a medium bowl until well combined. Work in 4 tablespoons (2 ounces; 57g) of the butter using fingertips until mixture is well combined and resembles wet sand. Set aside.

In a large bowl with an electric mixer, beat granulated sugar and remaining ⅔ cup butter on medium speed until fluffy, 3 to 4 minutes, stopping to scrape down sides as needed. Add eggs, one at a time, beating until combined after each addition, about 30 seconds.

In a medium bowl, whisk together salt, baking powder, baking soda, remaining 1 ¾ cups flour, and remaining 1 teaspoon cinnamon in a medium bowl. In a small measuring cup with a spout, whisk together sour cream and cider until well combined.

Stir flour mixture into butter mixture in 3 additions, alternating with sour cream mixture, until flour mixture is moistened but large flour streaks remain. Fold cooled apples into batter until evenly dispersed and batter is just combined. (Do not overmix.)

Spoon 2 tablespoons batter into bottom of each paper liner. Sprinkle 1 packed teaspoon brown sugar mixture evenly over batter in each muffin liner; top with remaining batter (about 3 tablespoons per well) spreading into an even layer. (Liners will be very full.)

Stir oats into remaining brown sugar mixture; sprinkle evenly over batter (about 2 packed teaspoons per muffin), and press gently to adhere.

Bake for 5 minutes. Reduce oven temperature to 375°F and continue baking until a wooden pick inserted in center comes out clean, 12 to 18 minutes, rotating pan halfway. Let cool in pan 10 minutes.

Carefully remove from pan and serve warm or at room temperature, drizzling with reserved strained apple juices from step 2 just before serving.