Remove the silverskin. Use a sharp knife to trim the pork tenderloin of any visible silverskin. To do this, gently slide the tip of a knife underneath the silverskin tilted up, then carefully slide the knife while pulling the membrane up, being careful to remove as little meat as possible. Discard.
Butterfly the tenderloin. Place the trimmed tenderloin on a cutting board, then hold the knife so that it is parallel to the length of the pork. Make a shallow cut from top to bottom, about 1-inch thick. Repeat 1-2 more times, being careful not to cut through the tenderloin, until it lays completely open like a book. Season the inside with ½ tsp Kosher salt.
Make the filling. Place 1 Tbsp olive oil and 1 Tbsp butter in a skillet over medium-high heat. Add red onion and shiitake mushrooms. Cook for 5-7 minutes, stirring occasionally, or until mushrooms begin to brown on both sides. Add roasted red pepper, spinach, 1 Tbsp finely chopped rosemary, lemon zest, and ½ tsp Kosher salt, then cook just until spinach has wilted. Remove from the heat.
Prepare the tenderloin. Add a thin layer of fontina on the pork tenderloin, leaving the outer ½-inch clear. Spoon the mushroom and spinach mixture down the center of the butterflied pork.
Roll and tie the tenderloin. Carefully roll the tenderloin into a cylinder, trying your best to completely enclose the filling. Use a piece of kitchen twine about 5x the length of the tenderloin to secure together. Season the outside of the pork on all sides with ½ tsp Kosher salt and ¼ tsp black pepper.
Prepare the grill. Preheat a gas grill for indirect cooking. On a three burner grill, you'll turn two burners on to medium-high, while the third burner remains off. Preheat until grill is 400°F/205°C.
Grill the tenderloin. Place pork over indirect heat, the cook turning every 5 minutes until an internal temperature of 145°F/63°C has been reached. This will take about 25 minutes. Allow the meat to rest for 10 minutes before slicing and serving immediately.