Oven Preparation: Initiate by preheating the oven to a temperature of 425°F. Subsequently, apply a thin layer of butter to an 8x8-inch baking dish to prevent sticking.

Preparation of Onion-Garlic Base: In a sizeable pot set over medium heat, melt the designated butter. Introduce the minced onion, allowing it to cook until it achieves a soft and translucent appearance, approximately requiring 5 minutes. Incorporate the garlic, and season the mixture with the specified quantities of salt and pepper. Allow this to cook for an additional 30 seconds.

Integration of Liquids and Potatoes: Into the pot, introduce the chicken stock, heavy cream, and previously sliced potatoes. Ensure to bring this mixture to a gentle simmer. It is imperative to avoid reaching a boiling point. Seal the pot with a lid and diminish the heat to medium-low, permitting the potatoes to cook until they reach a tender state, typically spanning 15-20 minutes. Taste and make seasoning adjustments if necessary.

Assembling for Baking: Transfer the prepared mixture to the pre-buttered baking dish. Evenly sprinkle the cheddar, provolone, and Parmesan cheeses atop the mixture.

Baking Process: Position the dish in the preheated oven, allowing it to bake until the mixture is effervescent and the cheese has achieved a molten state. This typically requires 10-15 minutes.

Serving: Once adequately baked, extract from the oven. For an enhanced aesthetic appeal, garnish with freshly chopped parsley. Serve while hot.