Step 1Heat the oven to 350 degrees. In a large, heavy sauté pan, melt the butter over medium heat until foamy. Add the carrots (this can be done in batches if necessary), and season with salt to taste. Very slowly cook the carrots, shaking the pan often, until the butter solids start to brown and the butter coats the carrots, and the carrots begin to color, about 45 minutes. Remove the carrots to a rimmed baking sheet and continue to roast until tender, about 30 minutes, depending on the carrots’ size.
Step 1In a small, heavy pot, combine the onion and butter over low heat. Cover and cook, stirring occasionally, until the onions are softened and translucent, 30 to 40 minutes. Stir in the sliced peach, cover and continue to cook just until the peach begins to release its liquid. If the mixture at any time becomes dry, add a little water to moisten.
Step 2Remove the mixture to a blender and purée, seasoning with 1/4 teaspoon rice wine vinegar and a generous 1/4 teaspoon salt, or to taste. This makes about 11/2 cups soubise.
Step 1In a skillet, brown the butter over medium heat. Add the carrots and toss until evenly heated and coated with butter.
Step 2In a separate skillet, heat the clarified butter over medium-high heat until hot. Add the chanterelles and sauté until softened, 6 to 8 minutes, tossing frequently. Remove from heat and toss in a pinch of salt.
Step 3Meanwhile, spoon 2 to 3 tablespoons soubise on each serving plate.
Step 4When the carrots are hot, add the vadouvan and toss to coat. Remove from heat and divide the carrots and mushrooms among the plates, topping each serving with dried peaches and nasturtium leaves. Serve immediately.