Preheat broiler, with rack 6 inches from heat source. Mix ricotta and nutmeg in a small bowl. Mix sugars, salt, and vanilla in a small bowl. Cut bananas diagonally 1/4 inch thick, keeping slices in order as you cut. Discard tips.
Spread 1/4 cup ricotta mixture in each of four 5-inch-wide by 1-inch-deep baking dishes. Fan 1 banana around edge of each dish, creating a circle. Gently press bananas into ricotta mixture. Using a pastry brush, lightly brush bananas with water, then sprinkle each with 2 tablespoons sugar mixture. Pat sugar into an even layer on top of bananas to prevent the sugar from clumping and burning.
Broil until sugar has caramelized and bananas have softened, 3 to 4 minutes. Serve immediately.
For an even faster dessert, broil sweetened banana, mango, or pear slices or halved pineapple by itself in an oven-proof dish. Top with ice cream, yogurt, or creme fraiche if desired. To broil fruit without ricotta, pat the sugar mixture into 1 pound of sliced fruit. Broil, rotating dish halfway through, until golden brown, 2 to 3 minutes.