Combine the oil, lemon juice, vinegar, chives, thyme, parsley, garlic, salt and pepper in a covered jar and shake well.
Preheat the oven to 400ºF.
Clean, trim and cut the celery root into two-inch cubes. Cut the potatoes, skin on, into two-inch pieces. Cut the carrots on the bias into two-inch pieces. Halve the shallots and garlic. Toss together in a large roasting pan with the oil, thyme, salt and pepper.
Roast for 45 minutes. Add the Brussels sprouts and roast for another 15 minutes.
Remove the thyme stems and toss with the lemon-herb vinaigrette.