For jammy tomatoes: Preheat oven to 250 degrees. Place tomatoes on a small baking sheet, drizzle with oil, and season with a generous pinch of salt & pepper. Roast for 2 to 2.5 hours, or until most of the tomatoes have burst and have browned in a few spots. Add tomatoes to a bowl and use a fork to smash into a "jam." Chill until ready to use.
For burgers: Dice mushrooms very finely. Add 1 Tbsp. oil and diced mushrooms to a cast iron skillet. Heat over medium heat for 9 to 10 min., or until the mushrooms have released their liquid and it has evaporated completely. Stir in 'Cue Glue and cook for 1 to 2 more min., or until mushrooms and sauce have formed a paste consistency. Place the mixture in a medium bowl and let cool for 15 min. Wipe cast iron skillet with a paper towel to remove any remaining bits. Add ground beef to mushroom mixture and use your hands to mix just until combined. Form 4 balls (if not cooking immediately, chill until ready to use).
Add remaining 1 Tbsp. oil to the cast iron skillet. Preheat skillet over high heat for 1 to 2 min., or until oil is very hot (but not smoking). Lower heat to medium-high. Season one side of the meat balls generously with salt & pepper. Place balls, seasoned side down, onto the hot pan (give them plenty of room, cook in batches of 1 to 2 burgers if needed). Use a large spatula to smash the balls into a thin patty—using the handle of a wooden spoon to help push the spatula down can be helpful. Season remaining side with salt & pepper. Cook for about 3 min., or until edges are browned and crisp. Flip, add a slice of cheese to each patty, and cover. Cook for 1 min., uncover, and cook for 1 to 2 more min., or until bottom is browned. Remove patties from heat and repeat process with remaining meat balls.
To serve: Smear a thick layer of jammy tomatoes onto the bottom bun. Add hamburger patty, top bun, and serve hot.