Boil the potatoes. Rinse the whole potatoes to remove any dirt. Place in a large pot and cover with cold water, about 1" above the potatoes. Bring the water to a boil, and then reduce heat to medium-low. Cover the pot with a lid and simmer the potatoes until fork-tender, about 20 to 25 minutes. (The potatoes should be done when the internal temperature is about 196-200℉.) Remove potatoes from the water and allow to cool slightly.
Prepare the pepper mixture. While the potatoes are boiling, add 3 dried guajillo peppers to a small saucepan. Add 1 cup of boiling water. Let the peppers sit in the boiling water for about 15 minutes until tender. Transfer the rehydrated peppers, about half of the liquid from the saucepan, and 1 chipotle pepper in adobo to a small blender cup. Blend until smooth.
Make the mashed potato filling. Remove the skins from the potatoes. Add the potatoes to a large mixing bowl with butter, half and half, salt, and pepper. Use a hand mixer to mix until smooth. Fold in the shredded cheese and the pepper mixture until well combined.
Assemble the potato tacos. Heat corn tortillas just until pliable, about 15-20 seconds. Place a spoonful of the mashed potato mixture on one half of a corn tortilla. Fold the tortilla in half to form a taco. Repeat until all tacos have been assembled.
Heat a skillet or griddle. Preheat large skillet or griddle to about 400-425℉. Add 3-4 tablespoons of high-heat cooking oil, enough to shallow fry the tacos.
Fry the potato tacos. Carefully place the folded tacos in the hot oil, a few at a time, and shallow fry until golden brown and crispy on both sides, about 2-3 minutes per side. Transfer to paper towels to drain off excess oil.
Serve and garnish. Garnish with shredded lettuce, diced tomatoes, sliced avocado, salsa, crumbled queso fresco or cotija cheese, or your favorite toppings. Serve hot.