In a small saucepan, combine potatoes with water to cover and season generously with salt. Bring to a boil and cook until just tender; drain and set aside.
Meanwhile, in a large skillet, cook chorizo over medium heat, breaking up large clumps and stirring, until browned, about 7 minutes. Add beans and oil and cook for 5 minutes, mashing gently with potato masher into chunky consistency.
Add drained potatoes to meat and beans mixture reduce the heat and cook until liquid is mostly evaporated, about 5 minutes. Season with salt, set aside, and keep warm.
Preheat the broiler. Split rolls, brush lightly with oil, and broil, cut side up, until light golden brown.
To assemble the sandwiches: On each bottom half, spread one-fourth of chorizo-potato mixture and top with cheese, onion, jalapeño, avocado, tomato, and lettuce. Top with other half of roll and cut in half.