For the brine:
In a large pot, stir together salt and water until salt dissolves. Stir in sugar, coriander, mustard, peppercorns, bay leaves, thyme, pepper flakes, and garlic. Add beef and submerge. Place a small plate on top of the beef to keep it under water. Cover pot and refrigerate 7 days.
For the corned beef:
Rinse brisket; discard brine. Place in a large pot. Add enough water to cover by 2 inches. Add onion, celery, and carrot, and bring to a boil. Reduce heat, cover, and simmer until very tender, 3 to 3-½ hours.
Transfer corned beef to a cutting board. Tent with foil, and let rest 30 minutes.
Heat oven to 400 degrees.
Discard remaining solids from broth and strain. Discard all but ½ cup broth.
In a large bowl, add cabbage and drizzle with olive oil to coat lightly. Arrange in a single layer on a foil-lined baking sheet. Add baby carrots to the bowl, drizzle with oil, and season with salt and pepper. Add to the foil-lined baking sheet and place in oven.
Repeat with Brussels sprouts, potatoes, turnips, and any other vegetables you choose to roast. Arrange them in single layers on a foil-lined baking sheets and place in oven. Try to place vegetables with similar cooking times on the same baking sheet.
Turn vegetables every 15 minutes or so until well-browned. Remove to a platter once they’re cooked and tent with foil.
Trim excess fat from beef. Slice thinly against grain, and transfer to platter. Serve with broth and mustard.