Position a rack in the center of the oven and preheat to 350°F. Grease an 8-inch square baking pan.
Brown the butter in a small saucepan over medium heat until light brown and nutty, about 6 to 8 minutes. Remove from heat and cool for about 10 minutes.
Mash the strawberries with 2 tablespoons of the sugar until juicy and lightly crushed. Set aside.
Whisk together the milk, lemon juice, egg, lemon zest, baking powder, salt, and vanilla in a large bowl until combined.
Stir in the remaining sugar.
Fold in the flour just until no dry streaks remain. Do not overmix.
Mix in the cooled brown butter until evenly incorporated.
Spread the batter evenly into the prepared pan.
Top with the mashed strawberries and all their juices.
Bake at 350°F for 10 minutes.
Reduce the oven to 325°F and bake about 45 minutes more, until golden and a toothpick in the center comes out clean.
Cool completely before slicing and serving.