Bring water to a boil in a large saucepan. Add green beans and shelled beans. Reduce the heat to maintain a lively simmer and cook, uncovered, stirring occasionally, until tender, 15-20 minutes. (The beans will not be completely submerged in the liquid.) Stir in onion, butter and salt. Remove from heat, cover and let stand until the butter is melted.
Meanwhile, preheat oven to 350 degrees F. Fill four 10-ounce ovenproof bowls with hot water to warm them while the oven preheats.
Empty the bowls and dry. Place them on a baking sheet. Evenly divide the bean mixture and butter sauce among the bowls. Using a large spoon make a 1/2-inch-deep well in the beans for each egg. Break an egg into each well.
Bake until the egg whites are just set, 15-20 minutes. Check after 15 minutes; if necessary, continue to bake, checking every minute or two, until the whites are set but the yolks are still liquid. Remove from the oven and let stand 5 minutes; the eggs will continue to cook a little bit. Serve with a generous grinding of pepper on top.
Serving size: 1 egg, 1 Cup beans each
Calories:204, Fat:11g, Sat Fat:5g, Mono Fat:3g, Cholesterol:201mg, Carbs:17g, Protein:11g, Fiber:6g, Sodium:379mg, Potassium:396mg