Mix the dressing ingredients together, and season to taste with salt and pepper. If the dressing is too thick for your preferences, thin it out with a little bit of cold water.
Up to two days in advance, season the chicken breast with the salt and chipotle powder.
Lightly brush the chicken breasts with oil, and grill the chicken over medium-high heat. Cook for a few minutes on the first side, until they easily release from the grill grates. Flip and continue cooking for a few more minutes, until the internal temperature reaches 160F. Remove from heat and top each with the shredded cheese. Cover to keep warm.
Lightly toast the buns.
Toss the lettuce with the dressing.
Assemble the sandwich: slather some dressing on the top and bottom buns, then place the lettuce on the bottom bun, followed by the cheesy chicken, chopped giardiniera, and the top bun.