Whisk together flour, baking powder, and salt in a large bowl. In another bowl, cream butter and sugar until pale. Add egg, vanilla, and lemon zest, mixing thoroughly. Gradually incorporate dry ingredients until a soft dough forms.

Wrap the dough in plastic wrap and refrigerate for 30 minutes. Preheat oven to 350°F (175°C).

Roll out the chilled dough to 1/4 inch thick, cut shapes with cookie cutters, and place on parchment-lined baking sheet. Bake for 10-12 minutes until edges are lightly golden. Cool completely on a wire rack.

Beat butter and powdered sugar together until smooth. Add lavender, heavy cream, and vanilla, mixing until creamy. Adjust consistency with additional cream if needed.

Spread frosting on the flat side of one cookie and sandwich with another. Garnish with lemon slices and lavender sprigs as desired. Chill briefly to set frosting before serving.