Preheat the oven to 350°F. In a large bowl, whisk together the sugar, buttermilk, butter, vanilla, and the juice of one lemon (about 3 tablespoons) until light and frothy. Stir in the eggs, one at a time.
In a separate bowl, combine the flour, baking soda, baking powder, and salt. Slowly fold the flour mixture into the wet ingredients until just combined. The batter should be thick, but still fairly fluffy.
Butter and flour-dust your desired baking tin. Pour the batter so that it fills each liner about 3/4 of the way full. Bake for 25-35 minutes if using cupcake tins or 35-45 minutes if using a cake tin. Bake until a toothpick inserted into the center of the cake comes out clean and the tops and sides of the cake are browned and slightly pulling away from the edges of the pan.
Cool the cake completely. When the cake is cooled, whisk together the remaining lemon juice, heavy cream, and powdered sugar until a thick glaze forms. Spread or drizzle the glaze across the tops of the cakes.
Serve room temperature or chilled. Store in the fridge for up to one week. Enjoy!