Heat the oven to 350°F and grease a 13x9x2-inch baking pan. Combine the flour, 2 cups sugar, baking powder, baking soda, nutmeg, and salt, then cut in the butter with a pastry blender or two knives. The dry mixture should look crumbly and uneven, with no large solid pieces of butter remaining.

Whisk the buttermilk, beaten eggs, and vanilla in a separate bowl, then stir them into the flour mixture just until the flour disappears. Fold in the mangoes, spread the thick batter in the pan, and sprinkle the surface with the combined 1/3 cup sugar and cinnamon. The batter should be thick and spoonable, holding soft mounds instead of pouring like pancake batter.

Bake for 40 to 45 minutes, then let the cake cool slightly before serving with sweetened whipped cream or Cool Whip, if desired. Check the center with a wooden pick, avoiding a molten mango pocket when testing. The top should be deep golden brown, the fruit may bubble lightly near the rim, and the pick should come out clean from the cake crumb.