In a medium bowl, combine flour, ground lavender buds, finely shredded lemon peel, and salt. In a large mixing bowl beat the 3/4 cup butter with an electric mixer on medium to high speed for 30 seconds. Add 1 cup powdered sugar. Beat until combined. Beat in as much of the flour mixture as you can with the mixer. Stir in any remaining flour mixture.
On a lightly floured surface, roll dough into a 10-inch-long log. Wrap log in plastic wrap or waxed paper. Chill for 2 to 24 hours or until firm.
Preheat oven to 350 degrees F. Cut roll into 1/4-inch slices. Place slices 2 inches apart on ungreased cookie sheet. Bake in preheated oven about 10 minutes or until edges are lightly browned. Transfer cookies to a wire rack and let cool.
For frosting, in a small bowl, combine the 1/4 cup butter, 1 cup powdered sugar, and the milk. Beat with a wooden spoon until smooth. If necessary, stir in additional milk, 1 teaspoon at a time, to make a frosting of spreading consistency. Spread frosting on tops of cookies and, if desired, garnish with additional lavender buds and thin strips of lemon peel. Makes 3 dozen cookies.
Place in single layer in an airtight container; cover. Store at room temperature for up to 3 days. Or freeze unfrosted cookies for up to 3 months; thaw and frost.