To make the vinaigrette: Season the quartered onions with salt and pepper, place in a dry ovenproof pan and roast in a hot oven until the onions start to blister and char, approximately 10-15 minutes depending on the temperature of your oven. Once the onions are cool enough to handle but still warm, place in the bowl of a food processor along with the vinegar and mustard. Pulse several times to begin breaking down the onions. With the machine running, drizzle in the olive oil in a thick stream. Leave the mixture chunky according to your taste. Stir in the shallot, adjust seasoning with salt and pepper and set aside.

Meanwhile, roast your carrots and greens. If your carrots are all roughly the same size, roast them whole. Toss them with half of the vegetable oil, season with salt and pepper, and place in oven. Cook until tender when pierced with a knife, 15-20 minutes depending on the temperature of your oven. Once the carrots are cool enough to handle, cut into bite-sized pieces and set aside. Toss the chicories with the remaining oil, season with salt and pepper, and roast in a hot oven until the greens begin to wilt and char slightly, 1-3 minutes.

In a large bowl, toss the greens with the carrots and dress with the onion vinaigrette to taste. Top with the shaved pecorino.