Step 1

Combine pork shoulder, spareribs, 6 quarts water, and 1 tablespoon salt in a large stockpot. Bring to a boil over high, skimming off and discarding foam from surface during first 10 minutes of cooking. Place onion, garlic, ginger, bay leaves, and oregano in center of a large piece of cheesecloth; gather edges of cheesecloth together, and secure with twine. Add to pot; reduce heat to medium-low, and gently simmer, uncovered, 1 hour and 30 minutes.

Step 2

Remove cheesecloth bundle from pot. Remove onion, garlic, and ginger; set aside. Discard bay leaves and oregano. Add hominy to pot; simmer over medium-low, uncovered, until rib bones can be easily removed from spareribs, about 1 hour and 30 minutes.

Step 3

While pork mixture simmers, split ancho and guajillo chiles; remove and discard stems and seeds. Place half of the chiles in a large, deep skillet over medium. Cook, turning occasionally, until toasted evenly on both sides, about 1 minute. Remove to a plate; repeat procedure with remaining chiles. Return all toasted chiles to skillet; add water just to cover chiles. Bring to a gentle simmer over medium; cook until chiles are soft and rehydrated, about 15 minutes. Remove from heat; cool slightly, about 15 minutes. Drain chiles, reserving cooking liquid. Taste cooking liquid. If it tastes bitter, discard. If it tastes faintly of raisins, reserve 1/2 cup.

Step 4

Transfer rehydrated chiles and either the reserved 1/2 cup chile cooking liquid or 1/2 cup water to a blender. Add onion, garlic, ginger, and 1 tablespoon salt. Process until smooth, adding splashes of stock from pork mixture in stockpot as needed to achieve consistency of applesauce, about 1 minute. Pour through a fine wire-mesh strainer into a bowl; discard solids.

Step 5

Heat oil in a 10-inch skillet over medium-high. Carefully pour strained chile mixture into skillet. (Mixture will splatter.) Cook, stirring often, until reduced by half and darkened in color, about 30 minutes. (Partially cover skillet to reduce splatters, if desired.) Remove from heat, and set aside.

Step 6

When pork mixture has finished simmering, remove spareribs from stockpot, and set aside until cool enough to handle, about 5 minutes. Remove and discard rib bones and tendons from spareribs. Chop meat into bite-size pieces, and return to stockpot. Stir in reserved chile sauce. Bring to a simmer over medium-low; cook until flavors meld, about 1 hour, skimming fat from surface, if desired. Stir in remaining 2 teaspoons salt.

Step 7

To serve, place tostada shells; lettuce; chicharrones, if using; Cotija, if using; radishes; avocados; and limes in separate bowls. Ladle soup evenly among 10 large bowls. For each serving, place a pinch of oregano between palms; rub over and into soup to release its aroma. Serve soup alongside bowls of garnishes for diners to customize their servings.