Put broth and chicken in a 2-3 quarter heavy saucepan and bring just to a simmer over moderate heat, uncovered. Turn chicken over and gently poach at a bare simmer, uncovered, until just cooked through, about 5 minutes more.
Transfer chicken to a cutting board. Pour broth through a fine-mesh sieve into a heatproof 2-cup measure and reserve for sauce.
Heat 2 tablespoons butter in a 4-5 quart wide heavy pot over moderately high heat until foam subsides, then cook peppers, stirring, until softened (do not brown), 6 to 8 minutes. Transfer peppers to a bowl and stir in 1/4 teaspoon salt and 1/4 teaspoon pepper.
Add onion and remaining 3 tablespoons butter to pot and cook over moderately low heat, stirring, until softened, 3-5 minutes. Add flour and remaining teaspoon salt and 1/4 teaspoon pepper and reduce heat to low, then cook, stirring, 2 minutes. Whisk in 3/4 cup broth, then all of cream and mushrooms, and simmer until mushrooms are tender, about 5 minutes.
Meanwhile, whisk together lemon juice, Sherry, and paprika in a small bowl. Whisk in 1/2 cup sauce, then stir mixture back into sauce remaining in pot. Cook over low heat, stirring (do not simmer, or sauce will curdle), until sauce is slightly thickened, about 2 minutes.
Remove from heat and reserve.
Cut chicken crosswise into 1/3-inch-thick slices and add along with peppers to sauce, then cook over low heat (do not simmer, or sauce will curdle), stirring occasionally, until chicken and peppers are just heated through. Add more broth to thin if desired.