In a large bowl, combine the ground beef, garlic, Old Bay seasoning, and parsley. Season to taste with salt and pepper. Form the mixture into 4 (1-inch-thick) patties. Season each burger with salt and pepper.
Heat a grill pan over medium-high heat, and brush lightly with canola oil. Grill the burgers over medium-high heat until a nice crust forms, about 4 minutes. Flip the burgers and continue cooking for another 3 minutes. Top each with 1 ounce of blue cheese and cook until the cheese softens and the burgers are medium-rare, 1-2 minutes. Remove the burgers to a plate and rest a few minutes before serving.
In a small bowl, whisk together the mustard and ketchup.
Serve on lightly toasted brioche buns with a dollop of Dijon-ketchup sauce and any combo of tomato, onion, and lettuce.
Do Ahead: Make the burger mixture and form the patties 1 day ahead; cover tightly with plastic wrap and keep in the refrigerator. Let sit out 30 minutes before grilling, to bring to room temperature.