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Heat the olive oil in a large skillet over medium heat. Add the sausage and brown for about 5-8 minutes, then remove with a slotted spoon to a paper towel covered plate.
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Add the onions with a pinch of salt to the oil. Cook, lightly caramelizing, for 10 minutes. Add the garlic and fresh thyme and toss for 2 minutes. Add another pinch of salt and the ground coriander.
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Add the mushrooms in 3 bunches, browning a little inbetween additions to avoid overcrowding the pan and steaming the mushrooms. Add a pinch of salt each time--mushrooms need salt! When the mushrooms have reduced a bit in size and are browning, after about 10 minutes total, add the apple slices and tomatoes. Toss and cook for 2 minutes.
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Add the browned sausage and toss again, and cook for another 3-5 minutes to allow flavors to meld. Taste for salt and pepper before serving and sprinkle with grated Parmigiano Reggiano.