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

In a bowl, beat butter, 1/4 cup confectioners' sugar, white sugar, lemon zest, and salt together with an electric mixer until fluffy.

Gradually sift flour into the batter, stirring in each addition.

Add 3/4 teaspoon lemon juice and gently knead until the dough holds together.

Press the dough into an 8x8-inch pan.

Bake in the preheated oven until shortbread is lightly browned around the edges, about 30 to 40 minutes.

Cool in the pan to lukewarm temperature and transfer to a wire rack to cool completely.

Warm milk in a saucepan over medium heat, stirring constantly, until milk begins to boil.

Remove from heat and add dried lavender flowers.

Cover the pan and steep until fragrant, about 6 to 10 minutes.

Strain the milk and discard the lavender flowers.

Gradually whisk 1 cup confectioners' sugar, 1 tablespoon at a time, alternating with 1 tablespoon Meyer lemon juice, until the glaze is smooth and opaque.

Spread the glaze over the cooled shortbread.

Refrigerate to allow glaze to harden, about 30 minutes.