Preheat oven to 350 degrees F. Bring about 8 cups water to a boil in a deep pot with 1½ tablespoons of salt. Carefully add potatoes to boiling water. Boil over medium-high heat for about 15 minutes. Check at 10 minutes. If they are tender but not firm then remove from the heat. You do not want to overcook the potatoes. Using a colander, strain the potatoes but do not rinse.
While the potatoes are cooking, crumble the chorizo into a cast-iron skillet or a casserole dish. Cover with foil and place in the preheated oven for about 15 to 20 minutes. Check after 10 minutes. The chorizo will release some oil but don’t discard it. (You can always cook the chorizo in a nonstick pan on the stovetop but I prefer this method because it makes less of a mess.)
Add the cooked potatoes to the cooked chorizo along with the remaining ½ tablespoon of salt to taste and gently combine. Do not stir too much or the potatoes will get mushy. Top with the shredded cheese. Place back in the oven, uncovered, just long enough for the cheese to melt.
Serve with tostadas, mini sopes, warm corn or flour tortillas, or use as in photo to fill gorditas.