Slice the nopales into 1⁄2-inch squares; you should have about 4 cups. In a bowl, toss the nopales with 2 tbsp. of the salt. Allow them to rest until the nopales have softened and released all their slime, about 30 minutes. Rinse well under cold water.
Heat the olive oil in a large pan over medium heat. When hot, add the garlic, chiles, 2 cups of the cured nopales, the oregano, cumin, pepper, and remaining 1 tsp. salt. Cook and stir well for 5 minutes. Add the vinegar, stir for 1 more minute, and remove from the heat.
Toss in the remaining 2 cups cured nopales and serve in a large bowl.