Apache plume (Fallugia paradoxa) has many more than ten pistils in each flower. These mature into dry fruits with long plume-like tails developed from the styles. So, the "feathery things" are feathery styles with one tiny achene (seed) attached. As you can imagine, these plumes assist with wind dispersal. F. paradoxa seeds are tiny (approx 550k/lb), have reasonable germination (+70%), and require no stratification. Other western U.S. shrubs with feathery styles include the closely related Stansbury cliffrose (Purshia stansburiana), and the Mountain mahoganies (Cercocarpus ledifolius, C. montanus, C. betuloides, and C. intricatus, which, along with Apache plume are all in the rose family, Rosaceae.