Step 1Heat the oven to 400 degrees.
Step 2Halve the squash lengthwise, then scoop out and discard the seeds and fibers. Place the halves on a baking sheet, cut side up, and season each half with one-fourth teaspoon salt. Place one-half tablespoon butter in the hollow of each half, then cover the sheet with foil and bake until the squash are tender, about 1 hour. Set aside to cool slightly, then scoop the flesh out of each half and discard the outer shell.
Step 3Brown the butter: Place 2 tablespoons in a large, heavy-bottom saute pan and cook over medium-high heat until the butter browns and has a nutty aroma, about 3 minutes. Be careful not to burn. Stir in the squash and sprinkle over the nutmeg. Reduce the heat to medium-low and cook for 5 minutes, stirring constantly to break the softened squash into a puree. Watch carefully so the squash does not burn on the bottom of the pan. Remove from heat and set aside to cool.
Step 4While the squash cools, cook the potatoes. Bring a small saucepan of lightly salted water to a boil. Stir in the potatoes, garlic and 1 sprig each of thyme and rosemary. Cook until the potatoes are just tender, about 8 minutes. Drain the potatoes, discarding the garlic and herbs and set aside in a large bowl.
Step 5In the same saucepan, combine the celery root with the cream and 1 1/2 teaspoons salt. Bring the mixture to a boil, then reduce to a simmer and cook until the celery root is tender, about 10 minutes. Strain out the celery root and place in the large bowl with the potatoes. Reserve one-fourth cup of the cream in a separate container.
Step 6Place 1 tablespoon butter in a large saute pan and brown over medium-high heat. Stir in the sage, carrots and parsnips and season with one-fourth teaspoon salt and a pinch of pepper. Cook over high heat, stirring frequently, until the vegetables are tender, 4 to 5 minutes. Remove the vegetables to the large bowl and set aside.
Step 7Clean the saute pan and brown 3 tablespoons butter over medium-high heat. Stir in the chopped thyme, apples, kohlrabi, leeks and turnips. Season with one-half teaspoon salt and a pinch each sugar and pepper. Increase the heat to high and cook, stirring frequently, until tender, 6 to 8 minutes. Remove the vegetables to the large bowl.
Step 8Clean the saute pan and melt the remaining one-half tablespoon butter over medium-high heat. Stir in the shallots and brown slightly. Stir in the spinach and cook just until wilted. Season with one-fourth teaspoon salt and a pinch of pepper. Remove from heat and add to the large bowl.
Step 9Add the chopped rosemary, diced brioche, reserved cream and creme fraiche to the vegetables and stir gently but thoroughly to combine. Taste and adjust the seasoning as desired.
Step 10Place the mixture in a baking dish so the filling reaches approximately one-half inch from the top of the dish. Using a rubber spatula, spread the butternut squash puree evenly over the top. Sprinkle the breadcrumbs evenly over the top.
Step 11Place the baking dish on a baking pan (to catch any drippings) and bake until the breadcrumbs are lightly browned, 20 to 25 minutes. Remove from heat and serve hot.