Remove the netting or strings on pork roast, keeping them intact to use again. Combine rosemary, 3 cloves of garlic, parsley, 3 teaspoons of olive oil, 1 tablespoon kosher salt, and 1 tablespoon freshly ground black pepper in a bowl. Slice open the pork roast and spread it with the oil and herb mixture. Roll up the pork and secure it with the netting or string. Rub outside of the roast with remaining 1 teaspoon olive oil and salt and pepper to taste.
Place the pork, fat side up, in a roasting pan or baking dish and pour in water. Bake in the preheated oven, basting occasionally, about 1 1/2 to 2 hours; add more water if necessary. An instant-read thermometer inserted into the meat should read at least 145 degrees F (63 degrees C). Allow the roast to rest 10 to 15 minutes. Reserve the pan juices.
While the pork is resting, heat 1 tablespoon olive oil in a frying pan over medium heat. Stir in 2 cloves chopped garlic and red pepper flakes; cook and stir until garlic is fragrant but not brown, about 3 minutes. Pour in chicken stock and bring it to a boil. Add spinach, 1 teaspoon salt, 1/2 teaspoon black pepper, and lemon juice. Reduce the heat to medium-low and cook until the spinach is wilted, stirring occasionally.
Cut the string on the roast and cut the pork into thin slices. Return the sliced pork to the reserved pan juices.
Slice the sandwich rolls and sprinkle each roll with shredded Manchego cheese. Add pork slices to each sandwich, top with spinach, and serve.