Place 1/3 cup butter in a 9-inch square pan and set in the preheating oven to melt, about 5 minutes.
Stir molasses, brown sugar, and ginger into the melted butter. Arrange pears cut-side down over mixture.
Sift flour, sugar, baking powder, and salt into a bowl. Add milk, 1/3 cup butter, and vanilla extract. Beat with an electric mixer until smooth, about 2 minutes. Beat in egg until combined, about 2 minutes. Pour batter over pears.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, 40 to 50 minutes. Cool in the pan for 5 minutes.
Run a knife around the edge of the pan; place a plate on top of the cake and flip carefully.