Place potato cubes in a large saucepan with enough generously salted water to cover; bring to a boil and cook until potatoes are barely fork-tender; drain and spread on a clean kitchen towel to dry.

Preheat oven to 350°F.

Sprinkle chicken on both sides with granulated garlic, granulated onion, and salt and pepper to taste. Place a large heavy ovenproof skillet over medium-high heat. Add cooking oil and swirl to coat pan. Add chicken, skin side down; place 1 tablespoon butter on each piece of chicken, transfer to the oven, and roast for 30 minutes. 

Heat 1 inch of vegetable oil in a large frying pan to 365°F. Add cooked potatoes and fry, turning occasionally, until golden brown on all sides. Drain on paper towels and keep warm.

Turn chicken over and continue to roast until browned on both sides and internal temperature reaches 160°F, about 10 minutes. Remove from oven, transfer chicken to a plate, and keep warm. Place skillet with pan juices over medium-high heat; add onion and cook, stirring, until softened and translucent, about 5 minutes. Add garlic and cook until fragrant, about 1 minute. Add mushrooms and green onions and cook, stirring occasionally, until mushrooms release their liquid and begin to brown, 6 to 8 minutes. Add ham, stir to coat, and deglaze pan with chicken stock. Reduce heat to medium and simmer until most of liquid has evaporated, about 5 minutes. Add reserved fried potatoes and combine; heat until warmed through. Taste and adjust seasoning. To serve, divide mushroom mixture among 4 serving plates; sprinkle with parsley. Top with a chicken breast and drizzle with Béarnaise Sauce.