Adjust oven rack to the middle position and preheat oven to 350°F (178°C). Line a rimmed baking sheet with parchment paper.
Pat chicken dry with paper towels and sprinkle with 1/4 teaspoon pepper. Wrap each breast with 2 slices prosciutto.
In a 12-inch (30-cm) oven-safe nonstick skillet over medium-high heat, warm 1 tablespoon oil until just smoking. Add chicken and cook until prosciutto is lightly browned, about 2 minutes per side. Transfer to prepared baking sheet and top each breast with 1 slice fontina.
Bake until chicken registers 160°F (71°C), 12 to 15 minutes. Transfer chicken to a serving platter, tent loosely with aluminum foil and let rest until the internal temperature reaches 165°F (74°C), about 5 minutes.
Meanwhile, in the same skillet over medium-high heat, warm the remaining 1 tablespoon olive oil until shimmering. Add asparagus and cook until just tender and spotty brown, about 4 minutes.
Stir in shallots, 1/4 teaspoon salt, and 1/8 teaspoon pepper and cook until shallots are lightly browned, about 2 minutes. Serve asparagus with chicken.