Heat 1 tablespoon olive oil in a large pot or Dutch oven over medium-high heat. Add the mushrooms, season with salt and pepper and cook, stirring occasionally, until tender and lightly browned, about 5 minutes. Add 1 more tablespoon olive oil and the sliced garlic; cook, stirring, until softened, about 30 seconds. Add the leeks, season generously with salt and cook, stirring, until they start softening, about 3 minutes. Add the chicken broth, rosemary sprig, 2 cups water and 3/4 teaspoon salt. Bring to a boil, reduce the heat to a steady simmer and cook until the vegetables are tender, about 10 minutes.
Meanwhile, preheat the broiler. Brush the remaining 2 tablespoons olive oil on both sides of the bread. Broil, turning, until toasted, 3 to 5 minutes. Rub the bread with the smashed garlic. Top with the ricotta and lemon zest; season with salt and pepper.
Stir the peas into the soup and heat through; season with salt and pepper. Divide the soup among bowls and serve with the toast. Sprinkle the soup and toast with the pecorino. Drizzle with olive oil.
Photograph by Ryan Dausch