Cut broccoli rabe crosswise into bite-size pieces. Transfer to a very large nonstick skillet (or two smaller ones) and add enough water to cover the bottom of the pan. Cover and cook over high heat until the broccoli rabe is bright green and most of the water evaporates, 2 to 3 minutes. Uncover, reduce heat to medium and cook until all water evaporates, 1 to 2 minutes. Add 1 tablespoon of oil and mushrooms, toss to coat, and cook until the mushrooms are soft and the broccoli rabe is tender, 2 to 3 minutes. Add garlic, thyme, pepper flakes, and salt, toss to coat, and cook 1 minute. Transfer the mixture to a plate and reserve the skillet.
Arrange 4 slices of bread on a work surface. Sprinkle half of the cheese over the bread slices, and then divide the broccoli rabe mixture over the cheese. Sprinkle with the remaining half of the cheese and top with the remaining bread. Compress slightly.
Wipe out the skillet and heat the remaining 1 tablespoon of oil in the skillet over medium-high heat. Add the sandwiches and cook until toasted on both sides and the cheese melts, 2 to 3 minutes per side. Cut in half on a diagonal and serve.
Tip: Add 1 ounce of sliced ham or prosciutto to each sandwich along with broccoli rabe filling. You can also omit the mushrooms if you’re not a fan of them.