To prepare the pork belly, preheat the oven to 250°F. 

In a small Dutch oven over medium-high heat, sear the pork belly on both sides, 3 to 5 minutes per side. Transfer to a plate. Pour off the excess fat from the pan.

Pour a little grapeseed oil in the pan and place over medium-high heat. Add the carrots and celery root and cook, stirring occasionally, until golden brown and caramelized, 8 to 10 minutes. 

Add the onions, garlic and apples and cook, stirring occasionally, until golden and caramelized. Add the tomato paste and cook, stirring occasionally, until deep brown in color, 5 to 6 minutes. Add the clove, star anise and cinnamon stick.

Add the Calvados and deglaze the pan, stirring to scrape up the browned bits. Simmer until the liquid is syrupy in texture. Add the pork belly and pour the warm stock over it; make sure there are no ingredients underneath the pork. Cover the pot, transfer to the oven and cook for 3 1/2 hours.

Remove the pot from the oven and let stand, with the lid on, for 1 hour to allow the pork belly to finish cooking. 

Uncover the pot, place a plate on top of the pork belly and add 20 pounds of weight to it. Let cool in the refrigerator overnight. 

The next morning, cut the pork belly into 1/2-inch-thick slices. Reserve the stock for glaze.

To prepare the apple pesto, in a bowl, stir all the ingredients together. Taste for texture, acidity, fat and earthiness. 

To prepare the caramelized shallots, heat a large, heavy-bottomed sauté pan over medium-high heat.

Add just enough grapeseed oil to create a shimmer in the pan. Add the shallots and do not stir for the first 5 minutes. Let them sear a little bit and then reduce the heat right after the initial sear. Stir the shallots every now and then from this point on. Do not let them burn or gain color too quickly; cooking low and slow is the best way to achieve pure onion sweetness.

Once the shallots are a deep brown color, transfer them to a baking sheet and let cool. Store in an airtight container the refrigerator until ready to serve. Serve at room temperature.

To serve, preheat the broiler to 500°F. 

In a sauté pan over medium-high heat, sear the pork belly slices on both sides until deep golden brown, about 1 1/2 minutes per side. Add the reserved pork stock and deglaze the pan, stirring to scrape up the browned bits. Simmer until the liquid is reduced to a syrupy consistency. Add the butter and rotate the pan steadily, allowing the butter to slowly emulsify. Remove from the heat. 

Place the English muffins on a baking sheet and toast under the broiler until golden brown on the edges and crispy underneath. 

Add a slice of pork belly and a large spoonful of sauce to each English muffin bottom. Top with the caramelized shallots. Grate the smoked provolone cheese over the shallots. Place under the broiler until golden brown. 

Top each with 1 tablespoon apple pesto, cover with the muffin tops and serve.