Preheat oven to 375°F.
Place strawberries, lemon juice, 2 tablespoons flour, and 1 1/2 tablespoons 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, remaining 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.
Bake for 17 to 20 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.