Heat the oil in a large skillet over medium-high heat. Working in batches to avoid overcrowding the skillet, add the chicken to the skillet; cook, stirring occasionally, until golden and no longer pink in the center, about 6 minutes. Transfer chicken to a plate.
Reduce the heat to medium. Add the mushrooms to the skillet; cook until the mushrooms release their juices and are golden and tender, about 6 minutes. (If the skillet becomes too dry, add 1 or 2 teaspoons of the chicken broth, or more as needed.)
Meanwhile, stir together the flour and 2 tablespoons of the milk in a measuring cup until the mixture is smooth and no lumps remain. Stir in the remaining milk until well blended.
Return the chicken to the skillet. Increase the heat to medium-high. Add the chicken broth to the skillet.
Stir flour mixture, salt, paprika and black pepper into skillet. Bring to boiling. Reduce heat to medium; simmer, stirring occasionally, until mixture thickens slightly, about 5 minutes. Serve with hot cooked rice.