Prep. Preheat oven to 350°F (175°C). Line a muffin tin with paper liners.
Wet Ingredients + Sugar. In a large bowl, whisk together the sourdough discard, oil, sugars, and eggs until smooth and well blended.
Dry Ingredients. In a separate bowl, whisk together the flour, baking powder, salt, and cinnamon.
Combine. Gradually add the dry ingredients to the wet ingredients, alternating with the milk. Begin and end with dry ingredients. Mix until just combined, being careful not to overmix. The batter should be thick but pourable. If it seems too dry, add a little more milk, a tablespoon at a time.
Add Apples. Gently fold in the diced apples.
Bake. Divide the batter evenly among the muffin cups. Bake for 22-27 minutes, or until a toothpick inserted into the center comes out clean. (Sourdough discard may require a slightly longer bake time.)
Cool. Let muffins cool in the pan for a few minutes, then transfer to a wire rack to cool completely.
Make the Glaze. While muffins cool, whisk together the glaze ingredients in a small bowl. If too thick, add milk 1 tablespoon at a time until the desired consistency is reached.
Drizzle Glaze on Top. Drizzle the glaze over the cooled muffins and allow to set before serving.