Either hand-sewing or machine, here's how you make one. Or, skip down to the bottom to see the toy in action with my cat Hambone.
First draw and cut out a donut (more properly a "doughnut," but that seems archaic, so I'm sticking with donut; I apologize, copy editors) from a sheet of scrap paper. Place this template on top of two layers of brown felt, and cut around the outer circle. Because these won't be exact, symmetrical shapes, mark a point on the same spot on each circle, so you can match them up again later. I used a pin, but later I switched to a white chalk pencil.