Serve this old-time cake with Brennan's vanilla ice cream or fresh whipped cream.
Preheat oven to 350° F. Butter a 10-inch pie plate. Sift flour, baking powder, and salt together into a medium bowl.
Put butter and 1 cup sugar in the bowl of an electric mixer fitted with the paddle attachment. Mix on medium-high speed until pale and fluffy, about 3 minutes. Reduce speed to medium-low; mix in egg, milk, and vanilla.
Reduce speed to low; gradually mix in flour mixture. Transfer batter to buttered pie plate. Arrange strawberries on top of batter, cut sides down and as close together as possible. Sprinkle remaining 2 tablespoons sugar over berries.
Bake cake 10 minutes. Reduce over temperature to 325° F, Bake until cake is golden brown and firm to the touch, about 1 hour. Let cool in pie plate on wire rack. Cut into wedges. Cake can be stored at room temperature, loosely covered, up to 2 days.