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.