Heat one tablespoon of oil in a 12-inch nonstick skillet over medium heat until it shimmers. Add the garlic and pepper flakes and cook until fragrant, about 1 minute.

Stir in the broccoli rabe, water, and salt. Cover and cook until the broccoli rabe is bright green and crisp-tender, 3 to 4 minutes. Remove the skillet from the heat, stir in the vinegar, then transfer the mixture to a bowl.

Heat the remaining two tablespoons of oil in the empty skillet over medium-high heat until shimmering. Add the mushrooms, cover, and cook, stirring occasionally, until the mushrooms release their liquid, 6 to 8 minutes.

Uncover the skillet and continue to cook until the moisture has evaporated and the mushrooms are golden brown, 6 to 8 minutes.

Stir the cooked broccoli rabe back into the skillet and season with salt and pepper to taste. Reduce the heat to low and shingle the cheese slices over the vegetables.

Cook until the cheese is completely melted, about 2 minutes, then fold the melted cheese thoroughly into the vegetable mixture. Divide the filling among the toasted rolls and serve warm.