Position racks in the top and bottom thirds of the oven; preheat to 400 degrees . In a medium bowl, whisk the flour, baking powder and salt. In a large bowl, using an electric mixer on medium-high, beat the sugar and 6 tbsp. butter until light and fluffy, stopping occasionally to scrape down the sides of the bowl, about 2 minutes. Add the egg, 2 tbsp. lemon zest, 2 tbsp. lemon juice and the vanilla. Beat until light and fluffy, stopping occasionally to scrape down the sides of the bowl, about 2 minutes. Add the dry ingredients; beat on low just to blend.
Using a 1-inch (2 tsp.) cookie scoop, scoop the dough and roll in colored sugar. Place onto two nonstick or parchment-lined baking sheets, spacing about 2 inches apart. Bake until the cookies spring back when pressed with a finger, rotating the sheets and reversing from the top to bottom racks halfway through baking, 7 to 9 minutes. Transfer the cookies to a wire rack and let cool. Repeat with the remaining dough on clean, cold baking sheets.
In a small bowl, using an electric mixer on medium, beat the remaining 1 stick butter, 1 tsp. zest, 1 tbsp. juice and the confectioners' sugar until smooth. Spread about 1 tbsp. frosting on the flat sides of half the cookies. Sandwich with the remaining cookies.