Dissolve the salt and sugar with the vinegar in a bowl or large Mason jar. Add the onion and toss around to coat. Let stand 15 minutes to soften.
Heat the olive oil in a sauté pan over medium heat. Cook the jalapeno, garlic and a pinch of salt a few minutes until softened and fragrant. Stir in the paprika and stir 30 seconds, then add the chickpeas and ¼ cup water. Cover and simmer 5 minutes.
Remove from the heat and mash the chickpeas coarsely with a potato masher or wooden spoon. Taste the chickpeas and season with more salt if needed.
Heat the tortillas directly over a medium gas flame on the stovetop until lightly browned, about 30 seconds per side (use a pair of tongs). Stack on a plate and cover with a towel to keep warm. If you don't have a gas stove, heat the tortillas one by one in a small heavy skillet over medium-high heat.
Make each taco with some of the the chickpeas, lettuce, cilantro, avocado, onion, cheese and cucumber. Squeeze lime juice over each taco and serve.
The chickpea filling makes enough for about 6 soft tacos. If you're feeding a crowd, double the recipe.