Pour 1 inch of canola oil into a large saucepan over medium-high heat.

Have ready a wire rack lined with paper towels to use for draining. Heat the oil to 350°F on a deep-fry thermometer and keep hot until ready to fry.

Meanwhile, place the chicken breasts between two sheets of plastic wrap and use a meat pounder or heavy skillet to gently pound them to a 1-inch thickness.

In a shallow dish, combine the flour, 0.5 teaspoon of the salt, and the pepper and stir to combine. Using a paper towel, pat the chicken dry and dredge in the flour mixture, shaking to remove any excess.

In a large skillet, heat 2 tablespoons of the butter and the 2 tablespoons canola oil over medium-high heat. Brown the chicken breasts on both sides, 5 to 6 minutes per side. Be careful not to crowd the pan; you may need to work in batches. Transfer the chicken to a platter and set aside, covered, with a foil tent to keep warm.

Add the chicken broth to the skillet, bring to a simmer, and cook for 2 to 3 minutes, whisking constantly to get all the browned bits from the bottom of the pan. Add the remaining 2 tablespoons butter, still whisking constantly.

Return the chicken to the pan and reduce the heat to low. Using a spoon, drizzle the pan sauce over the chicken and cook until the sauce has thickened, 4 to 5 minutes. Keep covered in foil or in a warm oven.

Use a vegetable peeler to cut the asparagus into ribbons to equal about 2 packed cups. Combine with the cornstarch in a large bowl and toss to coat.

Working in batches, fry the asparagus ribbons in the oil, tossing with a slotted spoon or spider occasionally, until crispy and starting to turn golden brown, 10 to 15 seconds. Remove to the paper towel-lined rack and sprinkle immediately with the remaining 0.125 teaspoon salt. Set aside.

To assemble, lay 1 slice of ham, 2 basil leaves, and 1 slice of Fontina on each chicken breast. Cook over medium heat for 2 to 3 minutes, using a spoon to drizzle the sauce on top to baste the chicken and melt the cheese.

Plate the chicken breasts, top each with a mound of asparagus ribbons, and drizzle each with 0.5 teaspoon balsamic glaze. Serve immediately, with the leftover sauce on the side.