Preheat the oven to 250 F and place a pan of water on the middle rack.
Bring a pot of water to a boil, salt heavily and add the beets, 3 thyme sprigs and the orange slices. Cook until tender, about 15 minutes. Drain well and discard the thyme sprigs and orange slices. Toss the beets with the olive oil and lemon juice and set aside.
Bring another pot of water to a boil, salt heavily and add the potatoes. Cook until tender, about 10 minutes. Quickly drain the potatoes and transfer to a large bowl filled with ice water until completely cooled. Drain the potatoes and dry well. Set aside.
Generously season the halibut with salt and transfer to an oiled rack. Place the rack over the pan of water in the oven. Cook until the fish is no longer translucent in the center and is slightly firm to the touch, 25 to 30 minutes, depending on the thickness of the filets.
In a large sauté pan over medium high heat, warm 1 tablespoon olive oil. Add the onions and lemons and let caramelize slightly, 3 to 5 minutes. Do not shake the pan.
When lemons begin to brown, add the mushrooms and salt lightly. Cook until the mushrooms have softened and begun to brown, about 5 minutes. Add the thyme and potatoes and salt to taste. Cook until the potatoes are heated through. Remove from the heat and add the mint.
Divide the potato-mushroom mixture equally among 4 plates. Place one halibut filet on top and spoon the beet mixture on top.
Adapted from Chef Barton Seaver, Hook, Washington, DC