Preheat oven to 375°.
Lightly spoon flour into a dry measuring cup; level with a knife. Sift together flour, 3/4 cup sugar, cocoa, baking powder, and salt over a large bowl. Add milk, 3 tablespoons butter, and vanilla, stirring until smooth. Set aside.
Combine 1 1/2 cups sugar and water in a small saucepan. Bring to a boil, stirring to dissolve sugar. Remove from heat.
Place a 10-inch cast-iron skillet in a 375° oven for 15 minutes. Place 1 tablespoon butter in preheated pan, swirling to evenly coat pan. Add batter, spreading evenly over pan. Pour water mixture slowly over batter; do not stir (mixture will bubble). Bake at 375° for 28 minutes or until cake is set. Let stand 10 minutes before serving.