Rinse beans with cold water; place in large saucepan. Add enough cold water to cover beans by 2 inches; let soak overnight. Drain. Return beans to same pan. Add enough cold water to cover beans by 3 to 4 inches.
Add celery stalk, carrot, and onion. Bring to boil. Reduce heat; simmer uncovered until beans are tender, about 1 hour. Remove from heat. Add 1 teaspoon coarse salt; let beans cool in cooking liquid. DO AHEAD: Can be made 2 days ahead. Cover and chill in cooking liquid. Before using, drain beans, reserving 1/4 cup cooking liquid.
Lightly butter 1-quart baking dish. Combine crab, cheddar cheese, mayonnaise, mustard, lemon juice, and Worcestershire sauce in medium bowl; chill.
Melt butter in large saucepan over medium heat. Add shallots; sauté until soft, about 3 minutes. Add chopped celery, garlic, and cayenne; stir 2 minutes. Add beans and 1/4 cup cooking liquid; cook until heated through, mashing beans coarsely, about 2 minutes. Add Crescenza cheese; stir until melted. Fold in crab mixture and 2 tablespoons parsley. Season with salt and pepper. Transfer to prepared baking dish. DO AHEAD: Can be made 1 day ahead. Chill crab brandade uncovered until cold, then cover and keep chilled.
Preheat oven to 400°F. Bake brandade uncovered until heated through and mixture begins to bubble at edges, about 30 minutes. Sprinkle with 2 tablespoons parsley. Serve warm with toasted baguette slices.
Ingredient tip: Crescenza cheese is a soft-ripened cows-milk cheese traditionally made in Italy. Look for it at your local cheese store, or order it online from or .
What to drink: Try a crisp California Chardonnay with the appetizers. We like the Cakebread Cellars 2007 Napa Valley Chardonnay ($39). Its bright and fruity, with notes of lemon zest and pear. The Napa Valley winery, which was founded in 1973, is currently run by four members of the Cakebread family.