Preheat the Oven and Prepare the Muffin Tin

1. Preheat the Oven: Preheat your oven to 350°F (175°C) and line a 12-cup muffin tin with paper liners or lightly grease the cups.
2. Prepare the Cinnamon Swirl: In a small bowl, combine the sugar, cinnamon, and melted butter for the cinnamon swirl. Set aside.

Make the Muffin Batter

1. Combine the Dry Ingredients: In a large bowl, whisk together the flour, sugar, baking powder, cinnamon, nutmeg, and salt.
2. Mix the Wet Ingredients: In a separate bowl, whisk together the pumpkin puree, milk, melted butter, egg, and vanilla extract.
3. Combine Wet and Dry Ingredients: Add the wet ingredients to the dry ingredients and stir until just combined. Do not overmix.

Assemble the Muffins

1. Fill the Muffin Tin: Spoon the muffin batter into the prepared muffin tin, filling each cup about 1/3 full.
2. Add the Cinnamon Swirl: Drop about 1 teaspoon of the cinnamon-sugar mixture into the center of each muffin. Use a toothpick or a small knife to swirl the cinnamon mixture into the batter, creating a marble effect.
3. Add More Batter: Spoon more batter into each muffin cup, filling the cups nearly to the top. Swirl the remaining cinnamon-sugar mixture into the batter.

Bake the Muffins

1. Bake: Place the muffin tin in the preheated oven and bake for 25-30 minutes, or until a toothpick inserted into the center of the muffins comes out clean.
2. Cool: Allow the muffins to cool in the tin for 5 minutes before transferring them to a wire rack to cool completely.

Make the Icing

1. Combine the Icing Ingredients: In a small bowl, whisk together the powdered sugar, milk, and vanilla extract until smooth.
2. Drizzle the Icing: Once the muffins have cooled, drizzle the icing over the top of each muffin.