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1
Place red potatoes and sweet potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 20 minutes. Drain. Add rice, milk, 2 tablespoons butter, and 1/2 teaspoon salt into the pot; mash until combined.
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2
Cover and refrigerate potato mixture until cool, about 15 minutes.
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3
Heat remaining 2 tablespoons butter and olive oil in a large skillet. Add mushrooms, garlic, fresh thyme, and fresh marjoram; cook and stir until mushrooms release their moisture and are tender, about 15 minutes. Pour in Madeira wine, scraping up browned bits from the bottom of the pan with a wooden spoon.
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4
Place 1/4 cup of the cooled potato mixture in the palm of 1 hand; flatten slightly. Place 1 tablespoon of the mushroom mixture in the center; form potato mixture into a ball around mushroom mixture using the back of a spoon. Repeat with remaining potato and mushroom mixtures.
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5
Combine 1/2 teaspoon salt, bread crumbs, Parmesan cheese, pepper, dried marjoram, and dried thyme in a shallow dish. Drop potato balls gently into the bread crumb mixture; roll around until coated. Press down to flatten potato balls into patties.
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6
Heat canola oil in a large skillet over medium heat. Cook patties 4 at a time until golden and crispy, about 4 minutes per side.
