For the chutney: Preheat the grill to medium heat. Brush the tomato halves with grapeseed oil and place on the grill. Grill for just 1 minute to get a char and soften the tomatoes. Chop the tomatoes and add to a bowl. Stir in the shallot, thyme, parsley, and garlic. Drizzle in just enough olive oil to moisten the mixture. Season with salt.
For the burger: In a bowl, blend the onions and beef and season with salt. Do not over-mix, this will cause the burgers to be too tightly bound. Next, portion into 2 balls, create a pocket in the center of each ball, and fill each ball with half the cheese. Season the outside of the balls and allow to rest for 5 to 10 minutes.
For the vegetables: In a bowl, mix the oil, vinegar, herbs, and garlic and add the vegetables. Allow the vegetables to marinate for 10 minutes. Grill over medium heat for 2 to 3 minutes each side. Increase the heat to medium-high.
After resting the burgers, pat each burger flat into a 1 by 6-inch patty. Grill over medium-high heat about 6 minutes on each side. The cheese should be still intact but not beginning to leak out.
Grill the brioche buns for just 1 minute. Stir the lemon juice into the mayonnaise.
Spread the lemon mayo on the bottom half of the buns, top with the burgers, some chutney, shredded romaine, and then the top of the buns. (Spoon the chutney onto the burger, leaving the liquid in the bowl.)
Serve alongside the grilled vegetables.