Wash the potatoes thoroughly under cold running water, scrubbing away any dirt. Peel approximately 60% of the skin from each potato, leaving about 40% for texture, color, and flavor. Cut the potatoes into even pieces, about 1 to 2 inches each, to ensure uniform cooking.

Place the potatoes in a large pot and cover with cold water by about 1 inch. Add a generous amount of salt to the water (about 1 tablespoon per quart). Bring to a boil over high heat. Reduce the heat to maintain a steady boil until the potatoes are tender and easily pierced with a fork, about 15 to 20 minutes.

Drain the potatoes in a colander and let them sit for 2 to 3 minutes to allow excess moisture to evaporate. This step is important for achieving the right texture.

Return the drained potatoes to the pot or transfer to a large mixing bowl. Mash the potatoes, leaving some small lumps for texture if preferred.

Add the butter and mix until melted and incorporated. Gradually add the heavy cream and mix gently until you achieve your preferred consistency. Season with salt. Taste and adjust the seasoning as needed.

Preheat the oven to 375°F.

Warm the vegetable oil in a large oven-safe skillet or Dutch oven over medium heat. Add the onions, carrots, mushrooms, garlic, and fresh thyme. Cook, stirring often, until the onions are translucent and the carrots are just tender, about 8 to 10 minutes.

Increase the heat slightly. Add the ground chuck and cook until well browned, breaking it up with a spoon as needed. Let the meat pick up color for a deeper flavor. Drain the excess fat from the pan if desired, and season the meat with salt and pepper.

Add the tomato paste and cook for 2 to 3 minutes, stirring to coat the meat and vegetables. Pour in the red wine, scraping up the browned bits from the bottom of the pan. Let the wine reduce slightly. Sprinkle the flour over the mixture and stir until well combined.

Gradually add the mushroom base mixed with water, stirring constantly to avoid lumps. Bring the mixture to a simmer and cook until it thickens into a gravy-like consistency, about 5 minutes.

Stir in the peas and zucchini. Remove the thyme sprigs and adjust the seasoning with salt and pepper.

Spread the beef filling evenly in the pan (or transfer to a casserole dish). Dollop the mashed potatoes over the top and spread into an even layer. If the pan is very full, place it on a foil-lined baking sheet to catch any drips. Bake until the potatoes are golden and the filling is bubbling around the edges, about 15 minutes.

Sprinkle the grated Parmesan cheese evenly over the potatoes and broil on high until golden brown. Watch carefully so the cheese doesn’t burn.

Let rest for 5 to 10 minutes before serving to allow the filling to set slightly.