1Sprinkle pork on all sides with fennel and season liberally with salt and pepper.
2 Lightly oil grill grate and preheatgrill to medium-high with an area for direct cooking and an area for indirect cooking. Grill pork over direct heat until well-seared, 4 to 5 minutes per side (trim any kitchen twine, if necessary, to avoid burning). Remove pork from the grill and let rest for 8 to 10 minutes. Meanwhile, grill lemons, cut sides down, over direct heat until well charred, 5 to 7 minutes. Remove lemons from the grill and set aside.
3 Return pork to the grill over indirect heat, cover, and cook until tender, 60 to 70 minutes.
4 Meanwhile, marinate the broccolini: In a large bowl, combine garlic, crushed red pepper, and oil. Add broccolini and toss to evenly coat. Set aside to marinate.
5 Then, make the applesauce: In a medium saucepan over medium-high heat, combine apples, cider vinegar, water, allspice, cinnamon, bay leaves and cloves. Cover and cook until steam starts to rise from the saucepan. Remove from the heat and set aside, covered, for 20 minutes.
6 Use a slotted spoon to transfer apples to the bowl of a food processor, leaving behind bay leaves and whole spices. Reserve remaining cooking liquid. Add about 1/3 of the butter and pulse to puree. Add about 1/3 more butter and puree. Add apple cooking liquid to desired consistency and remaining butter to taste. Season with salt. Cover to keep warm and set aside.
7 Remove pork from the grill and let rest for 5 to 10 minutes.
8 Finish the broccolini: While pork is resting, in a very large nonstick skillet over medium-high heat, add broccolini mixture and cook, stirring occasionally, until garlic is fragrant, about 1 minute. Add 1/4 cup water, cover, and cook 2 minutes. Uncover and cook, stirring occasionally, until broccolini is crisp-tender, 2 to 4 minutes. Season with salt. Cover to keep warm and set aside.
9 Cut pork into 1/2-inch slices. Spoon applesauce onto plates or a platter, top with pork, and arrange broccolini and grilled lemons alongside. Drizzle with additional olive oil, sprinkle with additional fennel and salt, and serve.