Preheat the oven to 425 degrees F.
On a small rimmed baking sheet, toss together the carrots, garlic, 2 tablespoons olive oil and 1/2 teaspoon salt. Cover with aluminum foil and roast until the carrots and garlic are both tender and soft, about 15 minutes. Remove from the oven and cool to room temperature.
Place the carrots, garlic and chickpeas in a food processor. Pulse to combine and break apart slightly. Add the lemon juice, 1/2 cup olive oil, cayenne and 1/4 teaspoon salt. Puree until smooth and an even pale orange.
Serve with root vegetable chips for dipping.
If using organic chickpeas, you may need to add 1/4 cup water just to loosen the mixture, as they tend to be a bit drier and tougher than regular canned beans.