Preheat oven to 425F. Grease or line a 12-cup muffin tin with paper liners. Set aside.
In a large bowl, whisk the sugar, eggs, oil, lemon zest, and vanilla together until combined.
3/4 cup granulated sugar, 2 large eggs, 1/4 cup neutral oil, 2 tablespoons fresh lemon zest, 1 teaspoon vanilla extract
Add the cottage cheese and milk, and whisk in until combined.
2/3 cup full-fat small curd cottage cheese, 1/3 cup milk
Stir in the flour, baking powder, and salt just until combined.
1 ½ cups sifted all-purpose flour, 2 ½ teaspoons baking powder, 1/2 teaspoon salt
Divide the batter evenly among the prepared muffin tins.
Sprinkle with coarse sugar.
Bake for 5 minutes.
Without removing the muffins, decrease the oven temperature to 350F and continue baking for about 10-15 minutes, or until a toothpick inserted in the center comes out clean.
Let the muffins cool in the pan for 5 minutes before transferring to a wire rack to cool completely.
Notes
Neutral oil: I like to use light olive oil, but any neutral oil, like canola oil, will work well.
Vanilla extract: For even more flavor, use lemon flavoring in place of the vanilla.
Cottage cheese: I recommend using full-fat cottage cheese for the moistest muffins. Also, using small-curd cottage cheese helps the curds to incorporate into the muffins.