In a large bowl, combine flour, lavender sugar and baking powder. Using a pastry blender or two knives, cut in margarine until mixture resembles coarse crumbs.
In a small bowl, whisk together egg, milk and lemon extract. Pour over flour mixture and stir just until a stiff dough forms (do not over mix).
Drop 1 tablespoonful (15 mL) about 2 inches (5 cm) apart on prepared baking sheets. Bake in preheated oven for 8 to 10 minutes or until bottoms are golden. Transfer to a wire rack and let cool completely.
In small bowl, combine icing sugar, milk, margarine and lemon extract to form a smooth, thin frosting.
Drizzle frosting over cooled cookies and sprinkle with lavender buds (if using).