Intelligent surfaces comprising of cost effective, nearly passive, and reconfigurable unit elements are lately gaining increasing interest due to their potential in enabling fully programmable wireless environments. They are envisioned to offer environmental intelligence for diverse communication objectives, when coated on various objects of the deployment area of interest. To achieve this overarching goal, the channels where the Reconfigurable Intelligent Surfaces (RISs) are involved need to be in principle estimated. However, this is a challenging task with the currently available hardware RIS architectures requiring lengthy training periods among the network nodes utilizing RIS-assisted wireless communication. In this paper, we present a novel RIS architecture comprising of any number of passive reflecting elements, a simple controller for their adjustable configuration, and a single Radio Frequency (RF) chain for baseband measurements. Capitalizing on this architecture and assuming sparse wireless channels in the beamspace domain, we present an alternating optimization approach for explicit estimation of the channel gains at the RIS elements attached to the single RF chain. Representative simulation results demonstrate the channel estimation accuracy and achievable end-to-end performance for various training lengths and numbers of reflecting unit elements.