Instructions

1. Place 1/2 stick unsalted butter, 2 large eggs, and 1/2 cup whole or 2% milk on the counter. Let sit at room temperature until the butter is softened, 1 to 1 1/2 hours.
2. Arrange a rack in the middle of the oven and heat the oven to 350°F. Line a standard 12-well muffin pan with paper liners.
3. Place 1 1/2 cups all-purpose flour, 1 teaspoon baking powder, and 1/4 teaspoon baking soda in a medium bowl. Whisk to combine and remove any lumps.
4. Finely grate the zest from 2 large lemons until you have 2 tablespoons zest. Juice 1 1/2 of the lemons until you have 5 tablespoons. Quarter the remaining lemon half and cut 12 thin slices from two quarters for topping the cupcakes if desired.
5. Place 1 cup granulated sugar, 1 tablespoon of the lemon zest, and 1/2 teaspoon kosher salt in the bowl of a stand mixer (or large bowl if using an electric hand mixer) and rub together with your fingers until combined and fragrant.
6. Add the butter and 1/4 cup vegetable oil to the lemon sugar. Beat with the paddle attachment on medium speed until fluffy, 3 to 5 minutes. Use a flexible spatula to scrape down the sides of the bowl. Beat in the eggs, one at a time, beating well after each addition.
7. Add half of the flour mixture and beat on the lowest speed until just combined, about 20 seconds. Add the milk, 4 tablespoons lemon juice and 2 teaspoons vanilla extract, and beat on the lowest speed until just combined, about 10 seconds. The batter will appear curdled at this point. Add the remaining flour mixture and beat on the lowest speed until just combined, about 20 seconds. Divide the batter evenly between the wells of the muffin pan (a heaping 1/4 cup each), filling almost to the top.
8. Bake until a tester inserted into the center of a cupcake comes out clean and the edges begin to brown, about 20 minutes.
9. Use a butter knife or mini offset spatula to help to remove cupcakes from the pan. Transfer to a wire rack and let cool completely, about 1 hour. Meanwhile, wash and dry the stand mixer bowl and paddle attachment (or electric hand mixer’s beaters and bowl). Place 1 1/2 sticks unsalted butter for the frosting in the bowl and let it sit at room temperature until softened.
10. Place 3 cups powdered sugar in a large bowl and whisk to remove any lumps.
11. Add the remaining 1 tablespoon lemon zest and 1/4 teaspoon kosher salt to the bowl with the butter. Beat with the paddle attachment on medium speed until smooth, about 20 seconds. Add the powdered sugar 1/2 cup at a time, beating on low speed and stopping to scrape down the sides of the bowl as needed.
12. Add 1 tablespoon whole or 2% milk and the remaining 1 tablespoon lemon juice. Beat on medium speed until the frosting is lightened in color and fluffy, 3 to 5 minutes more. Spoon the frosting onto the cupcakes, or transfer to a piping bag fitted with a tip and pipe onto the cupcakes. Garnish with the reserved lemon slices and sparkling or sanding sugar if desired.