Heat oven to 350 degrees F. Coat a 13x9-inch pan with cooking spray.
In a medium mixing bowl, sift together flour, baking powder, baking soda and salt. In a large mixing bowl, using a mixer on medium speed, beat together butter, sugar and vanilla until creamy - about 1-2 minutes. Beat in eggs, one at a time, until well combined and fluffy. Reduce mixer speed to low, and beat in flour mixture, alternating with milk until just combined. Spread batter into prepared pan. Bake 20-25 minutes, or until light golden brown and center of cake springs back when lightly touched. Cool on a wire rack.
To make the frosting, whisk together sugar, flour and milk in a small saucepan until well combined; bring to a boil, whisking frequently (mixture should thicken up). Transfer to a large bowl and chill in the refrigerator/freezer until room temperature and no longer hot.
Beat in butter, using a mixer on medium speed, until well combined and fluffy. Beat in vanilla and melted chocolate until well combined and fluffy - about 2-3 minutes. Spread frosting over cooled cake. Top with sprinkles.