Combine the strawberries, lemon juice, 2 tablespoons of the flour, and 1 1/2 tablespoons sugar into a small mixing bowl. Mix together until well coated. Place in refrigerator and set aside.
In a mixing bowl, combine together the remaining flour (sifted), remaining sugar, baking powder, baking soda, and salt. Incorporate the butter until the mixture becomes grainy.
Add the yogurt and vanilla extract. Mix until combined, do not over mix. Fold in the strawberry mixture and lemon zest. The mixture should be combined well.
Line a baking sheet with parchment and spoon on 2 tablespoons of dough at a time, about 1 inch apart.
Bake for 17 to 22 minutes or until the cookies are light brown on top. Remove from the oven and transfer cookies to a cooling rack. Allow cookies to cool completely.
While the cookies cool, prepare the glaze: Combine the powdered sugar, milk and vanilla into a small bowl and gently stir together with a fork until all of the lumps are gone. Drizzle glaze over cookies and serve or allow to set, about 30 minutes, before serving.