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Preheat oven to 375 degrees. In a large skillet, heat 1 tablespoon oil over medium. Add mushrooms and shallots and season with salt and pepper. Cook, stirring, until mushrooms are tender and browned, 10 minutes. Add remaining 2 tablespoons oil and the flour. Stir to combine and cook 1 minute.
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Add chicken broth, whisking constantly, and bring to a boil. Add chicken and rice and return to a simmer. Stir in peas, parsley, and sour cream; season with salt and pepper. Transfer mixture to an 8-inch square baking dish. Top with breadcrumbs.
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Bake until top is golden brown and mixture is warm throughout, 15 to 20 minutes.