Preheat the oven to 275°. In a small saucepan, cover the garlic cloves with water and bring to a boil. Cover and simmer over moderately low heat until the garlic is barely tender, about 10 minutes. Drain.
Meanwhile, in a bowl, cover the dried apricots with hot water and let stand until softened, about 10 minutes. Drain.
Open the pork roast, snipping off the strings if necessary, and set it on a work surface, fat side down. Season the inside of the roast with salt, black pepper and crushed red pepper. Arrange the garlic cloves and dried apricots in the center of the roast. Roll up the roast and tie at 1-inch intervals with kitchen string. Season the outside of the roast with salt, black pepper and crushed red pepper and set it in a roasting pan, fat side up. Roast the pork for 3 hours.
In a small bowl, combine the mirin with the soy sauce, honey, mustard and sesame oil. Brush the pork with half of the glaze and roast for 10 minutes. Brush the pork again with the remaining glaze and roast for about 10 minutes longer, until the pork is nicely lacquered and an instant-read thermometer inserted into the thickest part of the roast registers 165° to 170°.
Transfer the pork to a carving board and let rest for 15 minutes. Stir 1/4 cup of water into the pan juices. Pour the juices into a small saucepan and ladle off the fat. Cover the juices and keep hot.
Discard the string and carve the roast into thick slices. Serve the pork, passing the pan juices at the table.