Put the potatoes and parsnips in large pot and cover with water. Bring the water to a boil, over medium heat. Salt the water and cook the vegetable until tender, 12 to 15 minutes. Drain and return them to the hot pot. Grate in 3 to 4 tablespoons of onion, then add the spinach, butter, half of the milk and all of the cream. Season with salt, pepper and a little freshly grated nutmeg, to taste. Mash the mixture together and taste to adjust seasonings. Add the extra splash of milk if the potatoes are too tight. Cool to room temperature, then stir in the eggs and transfer the mixture to a casserole. Cover with cheese and sprinkle with paprika. Cool and chill for a make-ahead meal.
Preheat the oven to 400 degrees F.
Put the casserole on a baking sheet and put it in the lower third of the oven. Bake until hot and golden, about 35 to 40 minutes.
In a medium baking dish toss the mushrooms with the garlic, thyme and olive oil. Roast until dark and tender, about 30 minutes. Stir in the lemon juice and season with salt and pepper, to taste.
Serve the mushrooms alongside the casserole.