How to Make It
Preheat the oven to 400°. In a large, deep skillet, melt 4 tablespoons of the butter. Add the carrots and water. Cover and cook over moderately low heat, stirring occasionally, until the carrots are tender, about 30 minutes. Uncover, raise the heat to high and boil off any excess water. Season with salt and pepper.
Meanwhile, in a large skillet, melt 1 tablespoon of the butter. Add the broccoli rabe, cover and cook over moderately high heat until just tender, about 2 minutes. Season with salt and pepper and transfer to a bowl. Lightly beat 1 of the eggs and stir into the broccoli rabe.
Wipe out the skillet and melt the remaining 1 tablespoon of butter. Add the shiitake, season with salt and pepper and cook over moderately high heat, stirring, until tender, about 5 minutes.
Line an 8 1/2-by-4 1/2-inch metal loaf pan with foil. Butter the foil. In a large bowl, using a pastry cutter or 2 sharp knives, coarsely chop the carrots. Lightly beat the remaining 4 eggs and add them to the carrots with the shiitake, cheddar, 1 1/2 teaspoons of salt and 1/2 teaspoon of pepper. Spread half of the carrot mixture in the prepared loaf pan in an even layer. Top with the broccoli rabe mixture. Cover with the remaining carrot mixture, smoothing the surface.
Set the loaf pan in a large baking dish. Add enough hot water to the dish to reach halfway up the side of the loaf pan. Bake for about 1 hour and 15 minutes, or until the cake is firm throughout. Let cool for 10 to 15 minutes.
Unmold the carrot terrine onto a platter. Cut into 1-inch-thick slices and serve.