6 tbsp. unsalted butter,
plus more for ramekins
3/4 cup flour
6 1/4 cups milk
2 1/4 cups grated Parmesan cheese
Kosher salt, to taste
6 eggs, lightly beaten
7 tbsp. extra-virgin olive oil
10 oz. black trumpet mushrooms
4 large shallots, thinly sliced
8 oz. black salsify or Jerusalem
artichoke, peeled and cut diagonally
into 1/4"-thick slices
1 sprig fresh thyme
1 dried bay leaf
1 cup canola oil
1. Melt butter in a 3-qt. saucepan over medium-high heat. Add flour and cook, stirring constantly, until smooth and light brown, 3–4 minutes. Slowly whisk in 6 cups milk; cook, whisking occasionally, until thickened, 15–20 minutes. Whisk in 2 cups Parmesan until smooth and season with salt; remove pan from heat and let cool for 5 minutes. Whisk in eggs. Divide béchamel sauce into 3 equal portions; set aside.
2. Heat 2 tbsp. olive oil in a 12" skillet over high heat. Add mushrooms and cook until soft, about 4 minutes. Add half the shallots and cook, stirring occasionally, until any liquid evaporates and mushrooms turn golden brown, about 3 minutes. Reserve 1/4 cup mushroom mixture for garnish; set aside. Put remaining mushroom mixture into a blender with one-third of the reserved béchamel sauce; purée, transfer to a bowl, and set aside.
3. Heat remaining olive oil in a 2-qt. saucepan over medium-low heat. Add remaining shallots, salsify, thyme, and bay leaf and partially cover; cook, stirring occasionally, until salsify is tender but not browned, 10–12 minutes. Discard thyme and bay leaf and set aside 1/4 cup of the salsify mixture. Place remaining salsify and one-third of the reserved béchamel sauce in a blender; purée, transfer to a bowl, and set aside.
4. Heat canola oil in a 1-qt. saucepan over medium-high heat until a deep-fry thermometer registers 350°; add reserved salsify mixture and cook until browned and crisp, about 4 minutes. Using a slotted spoon, transfer salsify to paper towels; reserve for garnish.
5. Heat oven to 350°. Grease six 8-oz. ramekins with butter and transfer them to a 9" x 13" baking dish. Pour 1" boiling water into baking dish. Put 1/3 cup reserved mushroom purée into each ramekin. Transfer baking dish to oven and bake until mushroom sformato is just set, about 5 minutes. Remove baking dish from oven and pour 1/3 cup salsify purée into each ramekin. Return baking dish to oven and bake until completely set and a toothpick inserted into the middle of each sformato comes out clean, about 35 minutes. Transfer to a rack and let cool for 5 minutes.
6. To serve, run the blade of a small paring knife around edges of the ramekins and invert each sformato onto a small plate. Heat remaining béchamel sauce, milk, and Parmesan in a 1-qt. saucepan over medium heat until hot, whisk until smooth, and spoon sauce around each sformato. Serve garnished with reserved mushrooms and fried salsify.