Put the potatoes in a medium saucepan of salted water. Bring to a boil and simmer until almost tender, about 5 minutes. Drain.
In a large nonstick frying pan, heat 1 tablespoon of the oil over moderate heat. Add the onion, garlic, and mushrooms and cook, stirring occasionally, until the mushrooms are browned, about 6 minutes. Add the chicken, 1/2 teaspoon of the salt, the pepper, and the thyme. Sauté until the chicken is almost cooked through, 3 to 4 minutes. Remove the mixture from the pan.
Wipe out the frying pan and then heat the remaining 3 tablespoons oil over moderately high heat. Add the drained potatoes and let cook, without stirring, for 6 minutes. Add the remaining 1/2 teaspoon salt, stir the potatoes, and cook until well browned, about 4 minutes longer. Stir in the chicken and mushrooms, the cream, and the parsley. Cook until just heated through, 1 to 2 minutes longer.
Test-Kitchen Tip This is the perfect place to use leftover cooked turkey or chicken. Just toss it in at the end with the mushrooms.
This satisfying dish will go with different drinks depending on the time of day. For brunch, try a mimosa (orange juice and sparkling wine). For dinner, serve a light, young Bordeaux from the Médoc in France.
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