Place the peas in a large stockpot, and pour enough water to cover by 2 inches. Soak for at least 6 hours or overnight. Drain and rinse, and return to the pot. Add the onion, broth, and water; bring to a boil, reduce the heat, and simmer until peas are tender, about 30 minutes.
Meanwhile, place potatoes in a medium saucepan, and add cold water to cover by 1 inch. Simmer until potatoes are tender, about 25 minutes. Drain and let cool, and cut into quarters.
In a grill pan or under a broiler, cook the sausage, turning occasionally, until nicely browned and cooked through, 12 to 15 minutes. Cut into 2-inch pieces.
Melt butter in a large nonstick skillet over medium heat. Add garlic, and cook until golden. Add escarole; cook, stirring until tender, 8 to10 minutes.
Add potatoes and escarole to black-eyed peas, and cook over medium heat until mixture returns to a simmer. Divide among six shallow soup bowls, and arrange sausage links on top. Season with the salt and pepper.