Preheat oven to 350°F. Line cookie sheets with parchment paper; set aside. In a large bowl beat the butter with an electric mixer on medium to high speed for 30 seconds. Add granulated sugar and beat until fluffy, about 2 minutes. Add the egg yolk, lemon peel, and vanilla; mix well. Add the flour and 1/4 cup chopped strawberries and beat until combined. Mixture will be crumbly. Use your hands to work the mixture until it forms a smooth dough.
On a lightly floured surface, roll dough to 1/4-inch thickness. Use a 3-inch round cutter to cut dough circles. Cut centers of dough circles with a 3/4- to 1-inch round or scalloped cutter. Reroll and cut scraps. Place dough rings on prepared cookie sheets.
Bake for 10 minutes or until edges are light brown. Remove and cool on wire racks.
For icing, in a small bowl stir together the powdered sugar, strawberry puree, and lemon juice. Drizzle over cooled cookies. Let stand until icing sets.