Heat 1 tablespoon olive oil in a medium saucepan over medium heat until shimmering. Add garlic and cook until aromatic, about 1 minute. Add beans, cream, and ½ cup water. Bring to a simmer, then mash beans lightly with a potato masher. Add water until a loose, porridge-like consistency is reached. Stir in cheddar and parmesan cheese and season to taste with salt and pepper. Keep warm.
Heat 2 tablespoons olive oil in a large skillet over high heat until shimmering. Add spinach, season with salt and pepper, and cook, tossing and stirring frequently, until wilted and starting to crisp, about 4 minutes. Stir in lemon juice and transfer to a bowl.
Heat remaining tablespoon oil in skillet over medium-high heat until shimmering. Add eggs, season with salt and pepper, and cook until whites are set but yolks are still runny, about 3 minutes.
To serve, stir half of scallions in white beans and adjust consistency with water as necessary. Transfer to warm bowls, top with sautéed kale then with a fried eggs. Sprinkle with remaining scallions and grate some parmesan over the top. Serve immediately.
