Put the parsnips, potatoes, carrots, and rutabaga in a large pot. Cover with cold water and add a generous pinch of salt. Bring to a boil, then reduce to a simmer. Cook for about 20–25 minutes, or until all the vegetables are very tender.

While the vegetables cook, heat the olive oil in a large skillet over medium heat. Add the sausages and cook for 12–15 minutes, turning often, until browned and cooked through. Transfer to a plate and keep warm.

Drain the cooked vegetables well. Return them to the hot pot for 1–2 minutes to let excess moisture evaporate. Add the butter, warm milk, salt, and pepper. Mash until smooth or slightly chunky, depending on your preference. Taste and adjust seasoning.

In the same skillet or a saucepan, melt 1 tablespoon butter over medium heat. Add the sliced onion and cook for 8–10 minutes until soft and golden. Stir in the flour and cook for 1 minute. Gradually whisk in the chicken broth. Stir in the Dijon mustard and simmer for 3–5 minutes until thickened. Season with salt and pepper.

Spoon the parsnip and winter vegetable mash onto plates. Top with 2 sausages per person. Serve with onion gravy if desired.