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Bake potatoes in a 425-degree oven as directed. When tender, remove from the oven, allow to cool until you can handle them, peel and place in a bowl. Mash with a fork.
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While potatoes are baking, cook the farro. Combine with 3 cups water in a medium saucepan. Bring to a boil, add salt to taste and cover. Reduce heat and simmer 50 to 60 minutes, until grains are tender and beginning to splay. Drain well and set aside.
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Remove needles and leaves from rosemary and thyme sprigs and place in a large frying pan with 1 tablespoon of the olive oil. Add pepper and heat over medium-high heat, stirring occasionally, until the herbs begin to crackle. Add chopped peanuts and cook for 2 minutes, or until golden and fragrant, and stir in chopped mushrooms and about 3/4 teaspoon kosher salt (or to taste). Cook, stirring often, until mushrooms have released all of their liquid and there is no more liquid in the pan, about 8 minutes. Stir in wine and continue to cook, stirring, until there is again no liquid left in the pan. Remove from heat and transfer to the bowl with the potatoes.
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Heat another tablespoon of olive oil over medium heat in the pan and add shallots. Cook until tender and beginning to brown, 2 to 3 minutes. Remove from heat and scrape shallots into bowl with mushrooms and potatoes. Add farro and Parmesan and mash everything together. Add salt and pepper, taste and adjust seasonings.
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Take up a heaped 1/3 cup of the mixture and form into a ball (you can wet your hands to reduce sticking). Roll ball in panko or chickpea flour, then gently flatten into a patty. Set on a plate and continue with remaining burger mixture. Refrigerate uncovered for 1 hour or longer (the longer the better).
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When you’re ready to cook, place a rack over a sheet pan. Heat 2 tablespoons of oil in a 12-inch, heavy nonstick frying pan over high heat. Swirl the pan to coat with the hot oil. Lower heat to medium. Place 4 to 5 patties in the pan (do not crowd), and cook until well browned on one side, about 4 minutes. Turn and brown for about 4 more minutes. Remove to rack. Heat remaining oil in the pan and cook remaining patties. Keep patties warm in a low oven until ready to serve. Serve with a salad and your choice of toppings, such as the usual (ketchup, mustard, relish), or yogurt raita, garlic yogurt, or chutney.