Melt the butter in a large pot over medium heat. Add the fennel, shallots, ½ teaspoon salt, and ¼ teaspoon pepper and cook, stirring occasionally, until the vegetables are tender, 4 to 6 minutes.
Add the mussels and wine and simmer, covered, until the mussels open, 4 to 6 minutes. (Discard any mussels that remain closed.)
Meanwhile, combine the oil, vinegar, and ¼ teaspoon each salt and pepper in a large bowl. Add the greens and toss to combine.
Serve the mussels with the cooking liquid. Sprinkle with the fennel fronds and serve with the salad and bread.