Season chicken breasts with half of the salt and pepper.
Heat vegetable oil in a skillet over medium-high heat and cook chicken for 6 to 7 minutes per side until the internal temperature reaches 165°F (74°C).
Transfer chicken to a cutting board, allow to rest for 5 minutes, then slice into thin strips.
In a small whisking bowl, combine olive oil, lemon juice, Dijon mustard, honey, and the remaining salt and pepper to create the dressing.
In a large mixing bowl, combine the shredded Brussels sprouts, toasted almonds, and dried cranberries.
Pour the dressing over the Brussels sprout mixture and toss thoroughly to coat.
Add the sliced chicken and grated Parmesan cheese to the bowl.
Toss gently one final time and serve immediately.