Adjust oven rack to middle position and heat oven to 350 degrees. Pound thicker end of chicken breasts between 2 sheets of plastic to uniform ½-inch thickness. Pat chicken dry with paper towels and sprinkle with pepper. Slightly overlap 2 prosciutto slices on cutting board, lay 1 chicken breast in center, and fold prosciutto over chicken. Repeat with remaining 2 prosciutto slices and remaining chicken breast.
Heat 1 tablespoon oil in 12-inch nonstick ovenproof skillet over medium-high heat until just smoking. Brown chicken lightly, about 2 minutes per side. Transfer to cutting board and top with fontina.
Add asparagus to skillet. Using tongs, toss asparagus with 1 tablespoon oil. Arrange half of asparagus with tips pointed in 1 direction and remaining asparagus with tips pointed in opposite direction. Arrange chicken breasts on top of asparagus and bake until chicken registers 160 degrees and asparagus is tender and bright green, about 12 minutes.
Transfer chicken and asparagus to individual serving plates. Whisk remaining ¼ cup oil, pecans, vinegar, shallot, mustard, and thyme into juices left in skillet until thoroughly combined. Season with salt and pepper to taste. Drizzle chicken and asparagus with dressing and serve.