1. Place pork in a Dutch oven and add enough cold water to cover. Add half the onion, 1 tablespoon salt, peppercorns, whole cloves, bay leaf, and thyme; bring to a simmer over medium heat. Reduce heat to low, cover, and cook until pork is tender, about 3 hours. Remove from heat and let pork cool to room temperature in the liquid.

2. Meanwhile, melt lard in a medium saucepan over medium heat. Pour off and reserve all but two tablespoons fat from saucepan.

3. Return saucepan to medium heat and add remaining one-half onion; cook, stirring occasionally, until soft, about 4 minutes. Transfer onion to a large bowl.

4. Transfer pork to a large bowl with onion. Pour cooking liquid through a fine-mesh sieve and set aside. Add cloves, nutmeg, cinnamon, 1/2 teaspoon pepper, and allspice to bowl with onion and pork and using a hand-cranked meat grinder or food processor, grind or pulse pork mixture.

5. Add reserved melted lard and broth as needed to make mixture a spreadable consistency; season with salt and pepper. Transfer pate to a crock pot or bowl, pour 1/4-inch layer of reserved pork fat over top, and refrigerate 3 days before serving.