Stir together the butter, brown sugar, flour, oats, cinnamon, and salt in a small bowl to combine. Set aside.
Preheat the oven to 325°F. Line two standard muffin tins with paper liners.
Whisk together the flour, baking soda, and salt in a medium bowl. Set aside.
Beat the brown sugar and vegetable oil in the bowl of an electric mixer on medium speed for about 2 minutes, until light and fluffy. Add the egg, sour cream, and vanilla extract and mix until well combined. With the mixer on low speed, add the flour mixture mixture in three additions, blending until just combined. Fold in the apples.
Using an ice cream scoop, scoop the batter into the muffin cups. Distribute the topping evenly over the batter. Bake for about 30 minutes, until a cake tester comes out clean and the top is golden brown. Remove from the oven and let cool for 10 minutes. Transfer the muffins to a wire rack and let cool completely.