I do the same as Crafter2243 but I don't use any stabilizer as the felt provides it's own stiffness, so stabilizer isn't needed. If the initial color stitch doesn't work for attaching a backing , I'll hand sew, machine sew, or glue the backing.
It all depends on the design. I would do them on felt. If it finishes with a satin stitch I may add another piece underneath the frame before stitching it out.
If that is not possible I leave the design on the stabilizer, take it out of the frame, lay another piece behind it and sew over the edge with the sewing machine, Sometimes I may leave a opening so I can stuff them.