Place strawberries, lemon juice, 2 tablespoons flour, and 1 1/2 teaspoons sugar into a small mixing bowl and toss together until well coated. Place in refrigerator and set aside.
In a medium mixing bowl, sift together the remaining flour, sugar, baking powder, baking soda, and salt. Cut in the butter until a fine/even, grainy texture forms.
Stir in the Greek yogurt and vanilla and mix until just combined. Fold in the strawberry mixture and lemon zest and mix until well combined.
Line a baking sheet with parchment and spoon on 2 tablespoons of dough at a time, about 1 inch apart. The dough will be sticky so a 2 tablespoon ice cream scoop works best.
Bake for 13 to 15 minutes or until the cookies have barely started to brown on top. Remove from the oven and transfer cookies onto a cooling rack and allow to cool completely.
While the cookies cool, make the glaze: Place the sugar, milk and vanilla into a small bowl and gently stir together with a fork until no lumps remain. Drizzle glaze over cookies and serve or allow to set, about 30 minutes, before serving.
The cookies are sweet enough that the glaze is not necessary.