First, let me give credit where credit is due. I belong to several online appliqué groups. These are fabulous resources and opportunities to share with quilters all over the country and the world. Members post questions, generously provide tips and answers, and enjoy fellowship (I posted links to a few of these groups below for anyone that might be interested). Very recently, a member of one of these groups, Jenny Craddock, posted her method of making circles. With her permission I am posting her method...
"Iron two sheets of Wash Away Appliqué Sheets (C&T publishing. I've had mine for some time, so don't remember where I got it but probably my local quilt store)... cut your circles with a hole punch. I recently bought a 5/8", 3/4", and a 1". I already had a 1/2" and a 1/4" and a couple of smaller ones. Put a dot of glue on the circle and place on the fabric. Cut your circle with a seam allowance, do a running stitch around it and carefully pull together. The advertising for the sheets says "wash out for a softer end result or simply leave it in...it will soften over time." I plan to leave it in. " Jenny Craddock.
I did not have any of the Wash Away sheets available but I did have a very similar product from Sharon Schamber. It did NOT work! The paper punch was instantly jammed on the fibers. I tried freezer paper - didn't work. Two sheets of freezer paper ironed together - still did not make a clean cut. Then my husband suggested I use "real paper." Huh - sigh - he always thinks he understands this quilting stuff. So he brought over a few sheet of printer paper and an old manila file folder. Well, sure enough, he was right. But please don't tell him! The punches were perfect every time; even the 1/4 inch. After a little more experimenting, it turns out that only "crisp" paper works well. For most of the world this is not an issue. But, for soggy, south Florida, in the dead of summer, this is a big issue. I decided to stick with manila folders- they are cheap, just sturdy enough to provide a nice edge, and thin enough to leave in if I desire. I imagine some purists are horrified at the thought of leaving a piece of paper in their quilt. Well, many award winning appliqué artists admit to leaving freezer paper inside very small shapes. In my book, anything 5/8 inch or less is considered very small. Plus, museums are filled with quilts loaded with newspaper foundations, glues, adhesives, interfacing, etc. 
I was not in the mood to mark and cut 120 fabric circles and I did not want to use glue (glues, starches, etc. are not suitable for our climate because they tend to attract flora and fauna, i.e. molds and bugs). Instead I rotary cut 120 1-inch squares (double the size of the finished circle). This took another 20 minutes.
I then freehand stitched a circle (does NOT have to be perfect) along the edge of the square. In the example I used a contrasting thread so you can see it. I then quickly snipped the corners to remove bulk. I placed the manila folder circle in the center of the stitched circle and pulled the thread to gather the edges up over the circle.
I smoothed the edges and tied the thread off in the back. Viola - a finished fast and easy circle that can be stitched down either by hand or by machine.
OK - so if you are still fretting about the piece of paper left in your block, after the circle is stitched, you can, from the back, snip the background underneath the circle, and pull the paper out from there. You can also iron the circle, then snip the basting threads to pop out the paper. Re-iron and stitch down. But really - I don't see the need.
For a stuffed circle, place a little tuft of batting under the paper shape before gathering. It's give a very nice effect.
Now for the bonus - EXTRA skinny stems - thinner than 1/4". Again, I have to thank Jenny. She has everyone on our online group modifying our clover bias makers. Jenny's tip for making 3/16" and smaller stems is to take a 1/4" clover bias maker (the smallest one on the market) and use pliers to squeeze the opening together until it's only 3/16" wide. Then feed a scant 1/2" bias strip of fabric through the bias maker following the manufacturer's directions. A perfect 3/16" bias strip comes out the other end. Next I am going to try to modify a bias maker to make 1/8" bias strip. Yes, I know extra skinny bias stems are easy using the back basting (template free) method. But, this mehtod also goes very fast. 
Enjoy these tips, enjoy the summer, and enjoy your quilting!Best stitches,
Mercy in Miami
From start to finish, each circle only takes about a minute to prep - including cutting the template, rotary cutting the fabric, and stitching the gathering. In addition, because there is no glue, or starch, or iron involved my project just became that much more portable.
