Preheat the oven to 425 degrees F.
Toss the fennel with olive oil and season with salt and pepper. Place the fennel bulb on a baking sheet and roast for 45 minutes, or until tender and caramelized. Remove the fennel from the oven and cool. Julienne the fennel bulb.
Preheat the fryer to 350 degrees F.
Fry the potatoes for 2 to 3 minutes, or until crispy and golden brown. Remove from the fryer and drain on a paper-lined plate. Season with salt and pepper.
In a saute pan, heat the extra-virgin olive oil. When the oil is hot, saute the fennel, green beans, and garlic together for 1 minute. Season with salt and pepper.
In a mixing bowl, toss the Sauteed vegetables with the potato crisps. Serve the salad warm or at room temperature.