The chile paste. Place the chiles in a small bowl, cover with hot water, and let stand 30 minutes to rehydrate, stirring occasionally to ensure even soaking. Drain, reserving 2/3 cup of liquid, then transfer chiles and reserved liquid to a food processor or blender.
Pulverize the bay leaves in a spice grinder or a mortar, then add to the blender, along with the vinegar, onion, garlic, mixed herbs, allspice and cloves. Process to a smooth puree (adding a little more water if needed to keep the mixture moving through the blades); press through a medium-mesh strainer into a small bowl
Set a large (6-quart) pot with a lid (preferably a Dutch oven) over medium-high heat and add the oil or lard. When hot enough to make a drop of the puree really sizzle, add it all at once. Stir constantly as the puree sears concentrate and darken into a spicy-smelling paste, about 5 minutes. Remove from the heat and season with salt.
Seasoning and pot-roasting the meat. Turn on the oven to 325 degrees. If you are using pork shoulder or butt, cut it into slabs roughly 3 inches thick (try to get them all about the same thickness so they'll cook evenly); leave a picnic ham whole, but make 1-inch-deep incisions every few inches all over the meat. Lay the meat into the pot with the chile paste, then flip it over to cover with the chile (slathering with a spoon or spatula to give an even coating). Pour 1/2 cup water around the meat, cover tightly and place in the oven.
Baste the meat every 30 minutes with the liquid and rendered fat that accumulates around it. After about 2 1/2 hours (the fresh ham may need another 1/2 to 1 hour), the meat will be fork-tender and will have darkened to an appetizing and crusty, rich, red-brown. If all the liquid evaporates during the cooking, leaving only chile paste and fat, dribble a little more water into the pan so you can go on basting. If time allows, let the pork stand, covered, for 20 to 30 minutes to reabsorb juices before serving.
Serving the meat. Line a serving platter with the lettuce leaves. With the help of tongs, spatulas or meat forks, transfer the meat to the platter, then taste the pan juices and add a little more salt if necessary. Spoon the juices over the meat, then scatter the onion rings and radish slices over all, to create a riot of color and texture.