Melt 3 tablespoons of the butter in a large saucepan over medium heat. Add the spaghetti and cook, stirring, until browned, 5 to 7 minutes. Stir in the broth, tomato puree, black beans, chipotles, and ½ teaspoon each salt and pepper. Cook, stirring occasionally, until the spaghetti is tender and the sauce is thickened, about 15 minutes.
Meanwhile, heat the remaining 1 tablespoon of butter in a large nonstick skillet over medium heat. Crack the eggs into the pan and cook until the whites are set and the yolks are still slightly runny, 3 to 5 minutes.
Top the noodles with the fried eggs, sliced radishes, watercress, and Parmesan.