We consider a networked control system where a plant is connected to a remote controller via a shared network that allows only one user to transmit at a given time. At each transmission time, the controller decides between sampling one of the plant's sensors or transmitting control data to the plant. We tackle the problem of simultaneously designing a policy for scheduling decisions and a policy for control inputs so as to optimize a quadratic objective. Using the framework of dynamic programming, we propose a rollout strategy by which the scheduling and control decisions are determined at each transmission time as the ones that lead to optimal performance over a given horizon assuming that from then on controller and sensors transmit in a periodic order and the control law is a standard optimal law for periodic systems. We show that this rollout strategy results in a protocol where scheduling decisions are based on the state estimate and error covariance matrix of a Kalman estimator, and must be determined on-line. We contrast the solution to this problem with the solution to the seemingly similar sensor scheduling problem where optimal scheduling decisions can be determined off-line. We highlight how the protocol obtained from the rollout algorithm can be implemented in a distributed way in broadcast networks. Moreover, it follows by construction of rollout algorithms that our proposed scheduling method can outperform any periodic scheduling of transmissions.