Place fava beans in medium bowl. Pour enough water over to cover by 2 inches. Cover and let soak at room temperature overnight. Drain fava beans. Peel off outer shell and skin from beans and discard.
Place fava beans in large saucepan. Add 8 cups water, potato, onion, and carrot and bring to boil, skimming off any foam that rises to surface with large spoon. Reduce heat to medium; add pinch of salt and simmer uncovered until beans are very soft, stirring occasionally and adding more boiling water as needed to keep beans submerged, about 1 hour 45 minutes. Drain. Transfer bean mixture to processor and purée until almost smooth. Season to taste with salt and pepper. Transfer fava bean purée to bowl.
Heat 2 tablespoons oil in large skillet over medium-high heat. Add dandelion greens, garlic, and crushed red pepper; sauté until greens wilt, about 2 minutes. Season to taste with salt and freshly ground black pepper. Place greens atop fava bean purée. Drizzle with remaining 1 tablespoon oil; sprinkle with Parmesan cheese.
Test-Kitchen Tip: Keep a pot of boiling water on the stove so you can add water to beans as needed.
Bon Appétit, May 2010, by Donatella Arpaia