- In a saucepan of salted boiling water, cook the potatoes and celery until the potatoes are tender. Drain and set aside.
- In a large saucepan, bring the broth to a boil. Add the mussels. Cover and cook for 1 to 2 minutes or until mussels open. Drain and keep the cooking liquid for another use. Discard any closed mussels. Remove the mussels from their shells and place in a bowl. Refrigerate, if necessary.
- Place about 54 half shells on a baking sheet covered with crumpled aluminum foil or coarse salt to stabilize the shells. Set aside.
- In a small saucepan, soften the garlic in the butter. Add the flour and cook for about 1 minute, stirring constantly. Add the milk and pesto and bring to a boil, stirring with a whisk. Season with salt and pepper.
- With the rack in the upper third of the oven, preheat the oven’s broiler.
- Spoon the mussels and vegetables into the shells. Drizzle with the sauce. Bake for 2 to 3 minutes. Top with arugula and drizzle with oil. Serve immediately.