Place the 2 ½ lb Yukon Gold potatoes peeled and cut into 2-inch chunks chunks in a large pot and cover with cold, well-salted water by about 1 inch. Bring to a boil over high heat, then reduce to a steady simmer. Cook for 15–18 minutes, until a fork slides through a chunk with no resistance.

Drain the 2 ½ lb Yukon Gold potatoes peeled and cut into 2-inch chunks thoroughly in a colander and return them to the hot pot. Let them sit on the turned-off burner for 1–2 minutes, shaking the pot occasionally — this steams off excess moisture and gives you fluffier mash.

Add the 4 tbsp unsalted butter to the 2 ½ lb Yukon Gold potatoes peeled and cut into 2-inch chunks and mash until the 4 tbsp unsalted butter is fully melted and incorporated. Gradually pour in the ½ cup whole milk warmed (and ¼ cup heavy cream if using), mashing and stirring until smooth and creamy. Season with 1 tsp salt and ½ tsp white pepper. Cover and keep warm over the lowest heat setting, stirring occasionally.

While the 2 ½ lb Yukon Gold potatoes peeled and cut into 2-inch chunks cook, heat the 1 tbsp vegetable oil in a large skillet over medium heat. Add the sausages in a single layer — do not pierce them. Cook, turning every 3–4 minutes, for 16–20 minutes total, until deeply golden-brown all over and cooked through with no pink in the center. An internal temperature of 160°F confirms they're done.

Transfer the cooked sausages to a plate and tent loosely with aluminum foil to keep warm. Leave any rendered fat in the skillet — you'll use the same pan for the gravy.

In the same skillet (with sausage drippings), melt the 4 tbsp unsalted butter over medium heat. Add the 2 large yellow onions halved and thinly sliced halved and 1 tsp granulated sugar. Cook, stirring frequently, for 15–18 minutes until the 2 large yellow onions halved and thinly sliced are deeply softened and golden-caramelized. Don't rush this step — the slow caramelization is what gives the gravy its depth and sweetness.

Add the 2 cloves garlic minced and thyme, and stir for 1 minute until fragrant. Sprinkle the 1 tbsp all-purpose flour over the 2 large yellow onions halved and thinly sliced and stir well to coat, cooking for another 1–2 minutes to remove the raw flour taste.

Pour in the 1 tbsp Worcestershire sauce and 1 tsp Dijon mustard (if using), then gradually add the 2 cups beef stock low-sodium preferred, stirring constantly to prevent lumps. Bring to a simmer and cook for 5–8 minutes, stirring occasionally, until the gravy thickens to a glossy, pourable consistency. Taste and season with 1 tsp salt and black pepper.

Spoon a generous mound of mash onto each plate. Lay 2 sausages alongside or on top. Ladle the onion gravy liberally over everything. Scatter with fresh fresh flat-leaf parsley roughly chopped if desired and serve immediately.