Lemon and Lavender Glazed Lemon Shortbread

Preheat oven to 400 degrees F (200 degrees C).

Beat butter, 1/4 cup confectioners' sugar, white sugar, lemon zest, and salt together in a bowl with an electric mixer until fluffy. Gradually sift flour into batter, stirring in each addition; add 3/4 teaspoon lemon juice. Gently knead until dough holds together. Press dough into an 8x8-inch pan.

Bake in the preheated oven until shortbread is lightly browned around the edges, 30 to 40 minutes. Cool in pan to lukewarm temperature and transfer to a wire rack to cool completely.

Warm milk in a saucepan over medium heat; cook, stirring constantly, until milk begin to boil. Remove from heat and add lavender flowers. Cover pan and steep until fragrant, 6 to 10 minutes. Strain and discard flowers. Gradually whisk 1 cup confectioners' sugar, 1 tablespoon at a time, alternating with 1 tablespoon Meyer lemon juice, until glaze is smooth and opaque. Spread over cooled shortbread. Refrigerate to allow glaze to harden, about 30 minutes.