Peel the carrots. Cut in half or quarters longitudinally so that the fattest side is about the thickness of a nice fettuccini noodle. Then, use a vegetable peeler to slice the carrots into ribbons. Keep each color of carrot separate because you'll cook them separately to avoid the colors bleeding together.
Make the pesto using a food processor or blender. First add the garlic clove and chop up. Then, add the tarragon leaves, pine nuts, parmesan, white vinegar and a dash of salt and pepper. Add about half the olive oil and pulse until blended. Continue adding olive oil until mixed thoroughly.
In a small pot bring a couple of inches of salted water to a simmer. Add the white carrot ribbons and simmer for 2 or 3 minutes until al dente. Remove with tongs, then add the yellow carrots. Then, cook the red carrots and finish with the purple carrots.
Arrange the carrot fettuccini on a plate and drizzle with the tarragon pesto.