Heat a tablespoon of oil in a large nonstick skillet over medium-high heat until shimmering. Add the garlic and red pepper flakes and cook until fragrant. Stir in the broccoli rabe, water, and a pinch of salt, then cover and cook until the stems are tender, 3 to 5 minutes.

Remove the lid and continue cooking until the liquid evaporates. Stir in the balsamic vinegar, transfer the greens to a separate bowl, and set aside.

Wipe out the skillet and heat the remaining oil over medium-high heat. Add the mushrooms and cook, stirring occasionally, until they have released their juices and turned a deep golden brown, 8 to 10 minutes.

Stir the cooked broccoli rabe mixture into the skillet with the mushrooms to combine. Season the mixture with salt and pepper to taste.

Reduce the heat to medium and divide the vegetable mixture into four equal piles. Top each pile with the cheese slices, cover the skillet, and cook until the cheese is completely melted and gooey, about 1 minute.

Use a spatula to transfer each cheesy portion into the toasted sub rolls. Serve immediately while the cheese is still hot.