Preheat oven to 375 degrees F (190 degrees C).

Combine mangoes, water, and 1/2 cup white sugar in a saucepan over medium heat; simmer until sugar is dissolved, 5 to 6 minutes. Drain syrup from mangoes, reserving syrup.

Beat 2 cups sugar and butter together in a bowl using an electric mixer until smooth and creamy. Whisk flour, baking powder, and salt together in a separate bowl until combined. Slowly mix flour mixture, alternating with milk, into creamed butter mixture, beating well after each addition. Stir vanilla extract into batter; pour into a 9x13-inch baking dish.

Arrange mangoes on top of the batter and pour 2 cups reserved syrup over mangoes.

Bake in the preheated oven until a toothpick inserted in the center of the cobbler comes out clean, 40 to 45 minutes.