In a bowl, cream the butter and sugar until light and fluffy, 2-3 minutes. Beat in the egg, lemon zest, lemon juice, and vanilla.
In a separate bowl, whisk the flour, baking powder, and salt. Gradually mix the dry ingredients into the wet until just combined, then stir in the chopped basil.
Cover the dough and chill 30 minutes for easier handling.
Preheat the oven to 350°F and line a baking sheet with parchment. Scoop tablespoon-sized portions, roll into balls, and place 2 inches apart on the sheet.
Bake 10-12 minutes, until the edges are lightly golden. Cool on the sheet 2 minutes, then transfer to a rack.
Dust with powdered sugar or drizzle with lemon glaze, if desired.