In a large skillet with a lid, heat 2 tbsp. olive oil over medium-high. Add the tomatoes and garlic, then shake the skillet and season with salt and pepper. Cover and cook, shaking occasionally, until the tomatoes begin to burst, about 3 minutes. Uncover and coarsely mash the tomatoes. Cook until the sauce thickens slightly, about 1 minute. Stir in the tarragon and vinegar; remove from the heat.
Heat a large cast-iron skillet over medium-high.
In a medium bowl, combine the meat with the scallions, Dijon, salt, pepper and a drizzle of olive oil; form into four patties (thinner in the centers for even cooking). Cook the patties, turning once, until browned, about 8 minutes. During the last minute or so of cooking, top the patties with the cheese. Tent the skillet with foil to let the cheese melt. Build the burgers with the bun bottoms, lettuce, patties, tomato-tarragon sauce and bun tops.