Preheat oven to 300°. Combine beans, onion, celery, mushrooms, parsley sprigs, halved head of garlic, 1 Tbsp. salt, 3 Tbsp. oil, and 2 qt. water in a large Dutch oven. Bring to a simmer over medium-high heat, cover, and transfer to oven. Bake until beans are fully cooked, tender, and creamy through and through but as intact as humanly possible, about 1 hour, depending on the type, brand, and age of beans; start checking every 10 minutes after the first 45 minutes. (When checking beans for doneness, stir gently and taste at least 3 beans—it isn’t finished until they’re all tender!)
Using tongs, fish out aromatics and discard. Season with salt. Be generous! Under-seasoned beans are barely worth eating. Let sit on stovetop, uncovered, until ready to serve.
While beans are cooking, make your pistou and prep the vegetables and garnishes. Using your sharpest knife, finely chop mint and ¾ cup parsley leaves. (A dull knife will just mash your herbs and cause them to turn dark around the edges.) Transfer to a small bowl. Add ¾ cup oil, grated garlic, and 1 tsp. salt and stir to combine; set pistou aside.
Trim and wash radishes, then slice as thinly as possible into coins (a mandoline really helps here). Transfer radishes to a small bowl, cover with cold water, and chill until ready to use.
Wash asparagus and trim woody stems by bending each spear near the cut end until you find the place where it wants to break naturally. Cut off tips, then cut each tip in half lengthwise. Slice now-tipless stalks crosswise into thin coins. Toss asparagus coins and tips and peas in a medium bowl; set aside.
When you’re almost ready to serve the stew, return beans to a gentle simmer over medium heat, taking care not to stir too much—you don’t want to bust up those beans!
Generously drizzle oil into a large cast-iron skillet and heat over medium until shimmering. Working in two batches, fry bread slices until golden brown on both sides, about 2 minutes per side. Season with salt and transfer to a plate. (You may have to add more oil between batches, as the bread will soak some of it up.)
When beans are simmering, add reserved asparagus and peas and cook, stirring gently, until asparagus coins are barely cooked but still bright green and crunchy, about 2 minutes.
Drain reserved radishes. Bring Dutch oven full of stew directly to the table. Serve with fried bread, pistou, radishes, and lemon wedges.