Make sauce: Put jalapeño, garlic, mayonnaise, lime juice, and cilantro in a blender (add cilantro last). Or, put ingredients in a deep bowl or jar and use an immersion blender. Blend on high speed until completely smooth. Season to taste with salt and pepper.
Make slaw: Combine cabbage, shallot, cilantro, olive oil, and lime juice in a medium bowl. Season to taste with salt and pepper and toss well.
Spread 1 tbsp. sauce on each bun half. Layer bottom halves with slaw, followed by a tomato slice, a sprinkle of salt, a chicken cutlet, and a handful of potato straws. Drizzle with more sauce and add bun top. Serve immediately.