In a bowl, combine flour, ground lavender buds, finely shredded lemon peel, and salt. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, beat the butter on medium to high speed for 30 seconds. Add the confectioner's sugar. Beat until combined.
Beat in as much of the flour mixture as you can with the mixer. Use a wooden spoon to 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 ¼-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.
To make the frosting:
Combine the butter, confectioner's sugar and milk. Beat with a wooden spoon until smooth. If necessary, stir in additional milk until desired spreading consistency is achieved.
Spread frosting on tops of cookies, or fill and sandwich cookies. 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.