Season the chicken evenly on both sides with 1½ tablespoons salt. Heat 1 tablespoon olive oil in a shallow pan over medium-high heat. Add the chicken and sear for 3-4 minutes on each side. Remove chicken from the pan (it will not be fully cooked) and set aside. Wipe out the pan if it looks charred.
Melt the butter in the pan over low heat. Add the garlic and cook, stirring, until fragrant, about 1 minute, then whisk in flour. Pour in the chicken broth, whisking, then add French’s Dijon Mustard, ½ teaspoon salt, and ¼ teaspoon pepper, and whisk until smooth. Simmer sauce for 1-2 minutes, then stir in the heavy cream. Return the sauce to a simmer, place the chicken in the sauce, and cover the pan with a lid. Simmer until chicken is fully cooked through, 5–10 minutes.
Remove chicken from the pan. Stir the minced parsley and lemon zest into the sauce, then taste for seasoning. Slice the chicken. Remove the sauce from the heat.
Heat 1 tablespoon olive oil in a large skillet over medium heat. When the oil is hot, add the mushrooms and season with salt to taste. Saute mushrooms, stirring occasionally, until golden brown, 7–10 minutes. Remove mushrooms from the pan.
Heat remaining 1 tablespoon olive oil in the pan. Add asparagus and sauté, stirring occasionally, until tender, about 3 minutes. Season with salt to taste. Add the peas and cook 1 minute more.
Place the vegetables on a plate. Top with sliced chicken and drizzle with the dijon sauce. Garnish with chopped parsley.