Put the red onion in a small bowl and cover with cold water; set aside at least 15 minutes. Meanwhile, make the dressing: Combine half each of the cilantro and cheese in a food processor or blender; add the buttermilk, lime juice and about one-quarter of the sliced jalapenos and puree. Season with salt.
Toss the salad greens with the remaining cilantro and half of the dressing in a large bowl. Drain the red onion and squeeze dry. Serve the salad topped with the onion, chicken, jicama, radishes and the remaining sliced jalapenos, cheese and dressing. Serve with tortillas and lime wedges.