Put sausage grinding tool, large and medium cutting plates, grinder knife, and meat hopper in the freezer.
Cut pork shoulder into approximately 1-in. cubes and put in a very large bowl. Add wine and all seasonings to pork shoulder and toss to coat thoroughly, then spread on 2 rimmed baking sheets. Cut fat into 1-in. cubes and put in a medium bowl. Freeze both shoulder and fat until semifrozen, about 2 hours. (Keep meat semifrozen for duration of recipe; otherwise it won't grind easily).
Assemble grinder and attach to stand mixer following manufacturer's instructions, making sure the chilled grinding tool is fully inserted (you have to twist it in). Grind fat through large grinding plate, a few chunks at a time, into a medium stainless steel bowl. Fat should fall crisply and cleanly out of the grinder; if it smears, it's too warm (return fat to freezer).
Transfer half of pork shoulder to a large bowl; keep remaining half of shoulder frozen. Grind pork through large grinding plate, a few chunks at a time, back onto its baking sheet and return to freezer. Change to medium plate and grind remaining half of pork onto another baking sheet or large bowl.
Mix both grinds of pork together with fat, using two large forks and breaking up any clumps of fat.
Form a small patty from meat and fry in olive oil in a small pan over medium heat to test seasonings. Taste and adjust salt and spices accordingly. If you would like the flavors to mellow, cover bowl with plastic wrap and chill 24 hours before continuing; for a spicier sausage, proceed with the recipe.
Continue forming patties, or make sausages by stuffing meat into casings. To stuff meat into casings, put grinding mechanism, meat hopper, and ground meat back in freezer. Flush casings with warm water using a small funnel inserted in one end of casing (do this in a clean sink). Soak casings in warm water at least 30 minutes and up to 1 hour.
Follow manufacturer's instructions for assembling stuffer and attaching to mixer, using the medium-size horn and greasing it with butter or lard. Push casings all the way onto the horn in one long piece, as though you were putting on a stocking. Knot the end of the casing. Take out 1/4 of meat from freezer and keep rest in freezer as you work (if it warms up, it will gum up the machine and won't have as nice a texture when cooked).
Guide meat into casings so it packs tightly and firmly and let finished sausages coil into a bowl. As each reaches the length you like--about 6 to 8 inches and slim is standard for this chorizo--twist it into an individual sausage link or tie off with kitchen twine. Prick any air holes with a sharp toothpick.
Freeze sausages or patties at this point, or cook. For sausages: Grill over indirect 300° heat or pan-cook over medium heat until they're plump, browned, and shiny (an instant-read thermometer inserted lengthwise reads 145°). For patties: Grill or pan-cook until crisp-edged and browned.
* A good one-stop shopping source for the paprikas in this recipe is spanishtable.com. Pre-order back fat and hog casings from a butcher shop.
Fresh chorizo keeps, chilled, up to 3 days; frozen in 2 layers of resealable plastic bags, up to 3 months. Cooked chorizo keeps 1 week; frozen, up to 5 months.
Note: Nutritional analysis is per sausage.