If your butcher hasn't yet done it, trim any silver skin or excess fat off the pork roast. Place the roast on a sheet pan.
Make the rub: Place the coffee, sugar, paprika, salt, pepper and cumin in a small bowl and mix with your fingers. Sprinkle the rub over the pork on all sides, rubbing it into the meat. You can cook it right away, but the roast will have even more flavor if you let it cure in the refrigerator for 2 to 4 hours. Set up your grill for spit-roasting and preheat to medium-high (375 to 400 degrees). For a Kalamazoo Hybrid Fire Grill, I recommend removing the grill grates and starting a wood fire in the drawer. Move the fire/coals to the front of the drawer before mounting the spit in the grill so that the fire is offset from the meat.
Thread the pork roast lengthwise onto the rotisserie spit. Place on the rotisserie. Toss wet wood chips on the coals or add a foil smoking envelope to the fire to create smoke. Spit-roast the pork until sizzling and browned on the outside and cooked to your taste preference, 1-1/2 to 2 hours for medium (155 degrees).
Transfer the pork to a cutting board or platter and let rest for 15 minutes before carving and serving. (Note: Use a heavy cleaver to cut through the backbone.)
Meanwhile, make the Red-Eye Barbecue Sauce. Heat the pork drippings or butter in a cast iron skillet. Add the onion and ham and cook over medium-high heat until browned, 3 minutes, stirring with a wooden spoon. Stir in the flour and cook for 30 seconds. Stir in the coffee and boil until reduced to 3/4 cup, 3 minutes, scraping the bottom of the pan with the wooden spoon to dissolve any burnt on meatjuices. Stir in the tomato sauce, cream, Worcestershire and sugar, and simmer for 3 minutes. Add salt (you won't need much - the ham is salty already) and pepper and additional sugar if desired to taste. The gravy should be highly seasoned.
