Heat oil in a pan over medium-high heat. Add cherry tomatoes and garlic, shake pan and season with salt and pepper. Cover and shake pan occasionally until tomatoes begin to burst. Uncover and mash tomatoes; let thicken. Add vinegar, scallion greens and tarragon, and remove from heat.
Preheat a cast-iron skillet over medium-high heat.
Combine meat with scallion whites, salt, pepper, Dijon and drizzle of olive oil. Form four patties that are thinner at the center for even cooking. Cook patties 8 minutes turning once. Top with cheese in last minute or so, cover with foil tent to melt.
Place lettuce, a burger patty and some burst tomato sauce on roll bottoms and set tops in place.