In a large nonstick frying pan, heat 1 tablespoon of the oil over moderate heat. Season the chicken with 1/4 teaspoon salt and 1/8 teaspoon black pepper. Put the chicken in the pan and cook for 5 minutes. Turn and cook until just done, about 5 minutes longer. Remove. When the chicken is cool enough to handle, cut the breasts crosswise into 1/4-inch slices.
Heat 2 tablespoons of the oil in the same pan over moderately high heat. Put in the bell pepper, shiitakes, thyme, and 1/8 teaspoon salt. Cook, stirring frequently, until the mushrooms are golden brown, about 5 minutes. Add the garlic and cook, stirring, for 30 seconds longer. Remove from the heat.
In a large pot of boiling, salted water, cook the orzo until just done, about 12 minutes. Reserve 1/2 cup of the pasta water; drain the orzo.
In the frying pan, combine the reserved pasta water with the remaining 2 tablespoons oil, the goat cheese, cream, 3/4 teaspoon salt, and 1/4 teaspoon pepper. Bring almost to a simmer over moderate heat, stirring, until smooth, about 2 minutes. Add the orzo, chicken, and 3 tablespoons of the parsley and stir to combine. Serve topped with the remaining 1 tablespoon parsley.
Wine Recommendation: Sauvignon-blanc-based wines go particularly well with goat cheese, and a piquant Sancerre, with its grassy currant and gooseberry fruit, will make this dish sing.