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Heat your oven to 300°F. Mix beans, onion, celery, mushrooms, parsley sprigs, halved garlic, 1 Tbsp. salt, 3 Tbsp. oil, and 2 qt. water in a large Dutch oven. Simmer over medium-high heat, cover, and move to the oven. Bake until beans are tender and creamy, but still whole, about 1 hour. Check every 10 minutes after the first 45 minutes and stir gently. Ensure all beans are cooked by tasting at least three.
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Remove and discard aromatics with tongs. Add more salt to taste and let the beans rest, uncovered, on the stovetop.
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While the beans cook, prepare your pistou and vegetables. Finely chop the mint and ¾ cup parsley leaves. Mix them with ¾ cup oil, grated garlic, and 1 tsp. salt in a small bowl. Set your pistou aside.
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Thinly slice your radishes and soak them in cold water in a small bowl. Keep chilled.
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Trim your asparagus, snapping off the woody ends. Cut the tips in half lengthwise and slice the stalks crosswise into thin pieces. Mix the asparagus and peas in a medium bowl.
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Simmer the beans over medium heat as you prepare to serve the stew. Stir gently to keep the beans intact.
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Drizzle oil generously into a large cast-iron skillet and heat until it shimmers. Fry the bread slices in batches until golden brown on both sides, roughly 2 minutes per side. Season with salt and set aside.
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Add the asparagus and peas to the simmering beans and stir gently. Cook until the asparagus is bright green and still crunchy, about 2 minutes.
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Drain the radishes. Bring the Dutch oven to the table and serve the stew with the fried bread, pistou, radishes, lemon wedges, and grated horseradish on the side.