Heat one tablespoon of oil in a 12-inch nonstick skillet over medium heat until the oil shimmers. Add the garlic and pepper flakes and cook until the garlic is fragrant and just beginning to turn golden.

Stir in the broccoli rabe, water, and half the salt. Cover and steam until the broccoli rabe is bright green and crisp-tender.

Remove from the heat and stir in the vinegar. Transfer the greens to a medium bowl and set aside.

Wipe the skillet dry and heat the remaining oil over medium-high heat until shimmering. Add the mushrooms, cover, and cook until they release their liquid, stirring once or twice.

Uncover the skillet and continue to cook until all the liquid has evaporated and the mushrooms are well-browned and reduced in size.

Add the broccoli rabe mixture back into the skillet with the mushrooms and season with the remaining salt and black pepper. Reduce the heat to low and lay the cheese slices in an overlapping layer over the vegetables.

Cover and cook until the cheese is completely melted. Fold the melted cheese thoroughly into the vegetable mixture so it forms a cohesive, creamy filling.

Divide the mixture evenly among the toasted rolls and serve immediately while the filling is hot and the cheese is gooey.