How to Make It
In a small bowl, cover the dried morels with the hot water. Set aside until the morels are softened, about 15 minutes. Lift the morels out of the soaking liquid and rinse under water to remove any grit. Set aside 8 small morels for garnish and coarsely chop the rest. Reserve the morel soaking liquid.
In a small saucepan, melt 1 tablespoon of the butter. Add the chopped morels and half of the shallots and cook over moderately high heat until the shallots are lightly browned, about 4 minutes. Add the white wine and cook until evaporated, about 2 minutes. Add 1/2 cup of the chicken stock and the reserved morel soaking liquid, stopping when you reach the grit at the bottom of the bowl. Boil until reduced by half, about 5 minutes. Add the cream and simmer over moderately low heat until thickened, about 5 minutes. Season the morel cream with salt and cayenne and remove from the heat.
Transfer the morel cream to a blender and puree until smooth. Return the morel cream to the saucepan and keep warm. Rinse out the blender.
In a large saucepan, heat 2 tablespoons of the olive oil. Add the remaining sliced shallots and cook over moderate heat until they are softened, about 3 minutes. Add the remaining 3 cups of chicken stock and bring to a boil over high heat. Add all but 1/4 cup of the peas. Add 1 tablespoon of the chopped mint and simmer over moderate heat until the peas are just tender, about 4 minutes.
Working in batches, puree the pea soup in a blender and pass it through a coarse sieve set over a heatproof bowl, pressing on the solids with the back of a wooden spoon or a plastic spatula. Return the soup to the pot, reheat gently and season with salt and cayenne.
In a small skillet, melt the remaining 1 teaspoon of butter. Add the reserved whole morels and peas and season with salt. Cook over moderately high heat until hot, about 1 minute.
7. Ladle the pea soup into 4 shallow bowls. Swirl in the warm morel cream and scatter the whole morels and peas over the soup. Garnish the soup with the remaining chopped mint and the mint leaves and serve immediately.