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.
Wine Recommendation: 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.