In a medium pot, heat the vegetable oil until shimmery. Add cubed potatoes, and cook, stirring occasionally, until potatoes brown on all sides, 10 minutes or so. Remove, and drain. Add frozen peas to a small pot of boiling water, and cook just until water returns to boil for a minute. Drain, and set aside.
Season the chicken breasts on both sides with salt and pepper. In a large oven-proof skillet, melt butter over medium heat, and when sizzling, add the chicken breasts and sauté until nicely browned on both sides, 5 to 7 minutes a side. Place pan with chicken in oven, and bake 10 minutes or so, until chicken is cooked through (it should be opaque white, and juicy, but not pink in the center. If pinkish when you cut into it to examine, bake another 5 minutes). Remove chicken to plate and keep warm.
Place skillet back on medium heat (be careful: handle will be hot), add mushrooms and garlic and cook, stirring, until mushrooms brown slightly and render some liquid. Add potatoes, peas and half the parsley, and heat through. Place chicken breast in center of serving plate, and drape the vegetables over the chicken breasts attractively. Garnish with remaining parsley.