Bring 1 ½ cups water to a boil in a saucepan. Add rice and return to a boil. Reduce the heat to low, cover, and cook until rice is tender.

Meanwhile, place a steamer insert into a saucepan and fill with water to just below the bottom of the steamer. Bring water to a boil. Add whole zucchini, cover, and steam until tender, 2 to 6 minutes. Transfer to a cutting board and let cool slightly.

Preheat the oven's broiler. Grease a 9x13-inch baking dish. Slice two zucchini and set aside for garnish. Dice the remaining zucchini.

Heat ¼ cup butter and oil in a Dutch oven until butter is melted. Add rice and diced zucchini, then sauté until golden, stirring frequently. Stir in Parmesan and Cheddar until melted, then season with salt and pepper. Let cool slightly, then quickly stir in eggs. Pour into the prepared baking dish and sprinkle generously with bread crumbs. Arrange the sliced zucchini over the top and drizzle with 2 tablespoons melted butter.

Place under the preheated broiler until lightly browned and bubbly.