Preheat the oven to 375 degrees F.
In a medium bowl, mix the broccoli rabe purée, garlic, shallots, and the chile to combine.
Make a cut down the center of the pork tenderloin about 1 ½ inch deep (do not cut all the way through).
Spread the purée evenly in the center of the pork. (If you have extra, reserve it for spreading on toast!)
Sprinkle the mozzarella and Parmigiano-Reggiano evenly in the center of the pork, over the broccoli rabe purée. Wrap the pork back up to encase the filling. Tie the roast closed with butcher’s twine (alternatively, you can insert toothpicks along the seam to keep it closed instead).
Grease an oven safe large sauté pan with 1 tablespoon of olive oil. Transfer the pork to the sauté pan, and drizzle the remaining oil on top of it. Season with salt, pepper, and thyme.
Roast the tenderloin until a thermometer inserted into the fattest part of the tenderloin reads 140 degrees F, 25-30 minutes. Rest for 10-15 minutes before slicing and serving.