Place the potatoes in a large pot and cover with water. Add 1/2 teaspoon salt and bring to a boil. Cook for 15 minutes until the potatoes are fork tender.
While the potatoes cook, season the chicken breasts with the remaining salt and pepper. Melt 3 tablespoons butter in a large skillet over medium heat. Add the chicken and cook for 6 to 7 minutes per side until golden and cooked through. Remove the chicken to a plate.
Add the minced garlic to the same skillet and cook for 30 seconds. Add the asparagus and cook for 4 to 5 minutes until bright green and crisp tender. Return the chicken to the skillet and spoon the garlic butter over the top.
Drain the potatoes and return them to the pot. Add the remaining 2 tablespoons butter, milk, and heavy cream. Mash until smooth and creamy. Season with additional salt and pepper if needed.
Divide the mashed potatoes among four plates. Top each with one chicken breast and a portion of asparagus. Sprinkle with chopped parsley before serving.