Preheat the oven to 450-degrees F. Combine 4 minced garlic cloves, 3 tablespoons vegetable broth, 2 tablespoons olive oil and salt and black pepper, to taste, in a small personal blender. Blend until smooth.
Mix the potatoes, fennel slices and garlic-mixture in a medium bowl. Transfer the potatoes to a baking sheet. Bake until tender and golden, about 30 minutes.
Bring a medium pot of salted water to boil over medium heat. Add the shelled fava beans and cook for 3 minutes. Immediately transfer the cooked beans to a large bowl of ice water to cool. Using a slotted spoon, transfer the beans to a separate bowl. Add the spinach to the boiling water and cook for 1 minute.Transfer to the bowl of ice water to cool. Drain the spinach and squeeze out excess moisture. Transfer the spinach to a food processor and set aside.
Peel the blanched fava beans. Each bean is encased in a tough outer-skin that needs to be removed.
Add the basil, 2 minced garlic cloves, 1 tablespoon olive oil, 2 tablespoons vegetable broth and salt and black pepper to the food processor with the spinach. Process until smooth as possible and set the pesto aside.
Heat 1 teaspoon of oil in a medium skillet over medium heat. Add the peeled fava beans and saute until golden, about 3 minutes. Add the roasted potatoes and pesto. Mix well to combine and cook to reheat the potatoes and pesto, about 3 minutes. Taste and adjust seasoning. Serve the potatoes and beans with lemon wedges.