In a large, deep skillet, cook the bacon over moderate heat until crisp, about 4 minutes. Using a slotted spoon, transfer the bacon to paper towels to drain. Pour off the bacon fat in the skillet.
Add 2 tablespoons of the butter to the skillet and melt over moderately high heat. Add the radish greens, season with salt and pepper and cook, stirring, until wilted, about 3 minutes. Scrape the greens into a bowl.
Melt the remaining 4 tablespoons of butter in the skillet. Add the radishes, shallots and crisp bacon and cook over moderately high heat, stirring a few times, until the radishes are golden brown, about 6 minutes. Add the sugar and cook for 2 minutes, until dissolved. Add the orange juice and boil, stirring a few times, until the radishes are barely tender and the sauce is lightly thickened, about 2 minutes. Stir in the radish greens, season with salt and pepper and serve.