In a medium saucepan, cover the pearl barley or spelt with 2 inches of water. Bring to a boil, cover and cook over low heat until the farro is tender, about 25 minutes, add more water if necessary during cooking time. Drain.
Meanwhile, scrub your artichokes and put them in a pan and cover with water and a pinch of salt. Boil until the artichokes are tender, about10 minutes, then drain. Slice the sunchokes 1/4 inch thick.
Boil another pan of water (or use a steamer) and add the Tuscan kale, cooking until just tender, about 3-5 minutes. Drain the kale and let cool slightly. Squeeze out any excess liquid from the kale leaves and then coarsely chop them.
In a small frying pan, heat 2 tablespoons of the mixed oil. Add the red onion and a pinch of salt and cook over moderately low heat, stirring occasionally, until browned, about 12 minutes. Remove from the pan. Now add the butter and 2 tablespoons of the mixed oil. Add the artichokes in an even layer and cook over high heat until browned on the bottom, about 3 minutes. Turn the artichokes, reduce the heat to moderately high and continue cooking until starting to brown, about 2 minutes. Push the artichokes to the side of the pan. Add 1 more tablespoon of the oil and the mushrooms (if using). Season with salt and pepper and cook over moderately high heat until browned, 3 minutes. Then add the pearl barley or spelt, kale and onion and cook, stirring everything together until hot. Season with salt and pepper and serve.