Put oven rack in center position and preheat oven to 350°. Rinse pork and pat dry. Season all over with salt, pepper, thyme, rosemary, and sage. Set pork in a roasting pan and drizzle with olive oil. Cut top off garlic head so cloves are exposed and place in center of pork. Cover top of each bone with a small cap of foil to prevent blackening in oven. Pour 2 cups broth into pan.
Roast pork, basting every 30 minutes with accumulated pan drippings. (If pan begins to look dry, gradually add up to 2 cups additional broth.) Cook until a digital thermometer inserted horizontally through middle of roast into center of thickest part reads 145°, 1 1/2 to 2 1/2 hours (see Notes), and garlic is soft (if garlic isn't soft, put it in a small ovenproof dish and cook up to another 30 minutes). Transfer pork to a carving board, tent with foil, and let rest.
Meanwhile, pour pan drippings into a large measuring cup; let sit until fat rises to top. Spoon off 4 tbsp. fat and reserve; discard the rest. Measure 1 cup juices (add broth if necessary to make 1 cup).
Set roasting pan on top of stove so it spans 2 burners. Put reserved fat in pan and whisk in flour. Cook over medium-low heat, whisking, until flour develops a nutty aroma and is deeply browned. Gradually whisk in reserved juices and 3 cups chicken broth. With a wooden spoon, scrape up any browned bits; then resume whisking until gravy is smooth. Increase heat to high and bring to a boil, whisking. Boil until gravy is thick enough to coat the back of a metal spoon, 3 to 5 minutes. Add salt and pepper to taste; pour through a strainer into gravy boat.
Pop garlic cloves out of their skins into a small serving bowl. Remove string from roast and foil covers from bones. Serve with roasted garlic and gravy. Carve between bones with a sharp knife.
Note: Nutritional analysis is per serving.