One of the things I've done when I didn't have any mylar is to make a thick template using freezer paper. I iron 3 or 4 layers together and then cut my shape. You can do the running thread thing, spray starch and then remove the template.

I like the method where you sew backing fabric around the circle, then cut a slit in the backing fabric and turn it inside out. There are lots of ways to make applique circles that don't cost much of anything.

I absolutely love Karen Kay Buckley's perfect circles! They are heavy, and stand up to high heat. I tried making my own out of mylar, but they warped and melted. I also tried freezer paper, but when I made my circles I used starch and heat, and the freezer paper seemed to lose it's "edge" from the spray starch. I ordered Karen's circles, and love the way they turned out. I'm finishing up my version of a Country Whig Rose quilt that has 104 turned edge circles on it.

I use Karen Kay Buckley’s and love them

I use multiple layers of freezer paper to make my circle templates, and compass or protractor I forget which it is called has a pivot point and a place for a pencil. I do a lot of circles. I also do a reverse applique if the backing isn't going to be pieced. I use freezer paper for that too. Press it to the back of the fabric (background) cut a hole out of the middle of the circle spot leaving about 3/8 to 1/2 inch, notch it all the way around press the tabs back to form a folded edge, let it cool then remove the freezer paper, place it over the "circle" fabric use a little stick glue to hold things in place long enough to stitch around the edge. That way there is no "bulk" under the edge on your circle.

I use either Karen Kay Buckley circles or make my own using heat resistant template plastic. It is difficult to cut smooth edge on circles, especially the smaller ones, so I use a nail file to smooth out the little peaks that happen when you cut.
I make my fabric circle a little larger than my template and do a running stitch around. I then use a small paint brush to brush starch on the edges and then draw the edges up by pulling on the thread. Press with a hot iron, pop out the template when dry and perfect circle.

if you get Mylar, be sure it is heat proof. good luck on your circles. they can be fun.

I use empty cereal boxes, cut the size circle I want, usually determined by the size saucers,plates, etc, cover in foil. Cut your fabric 1/4" - 1/2" larger than the cir le you want to make, run a line of basting stitches around the outside of fabric. Once you have your basting done, place onto foil covered circle and draw up thread snugly and press. I leave basting thread in until partially sewn down. Has work well for me for years. Whatever way you choose, have fun with it!

Once I started using Appliquick tools and techniques I won't do applique any other way. I get perfect circles from 1/4" round to as big as I want. The tools are excellent. I use stabilizer instead of fusible for most of the applique.

https://www.youtube.com/watch?v=PoQdaN6hfUI

I save the cardboard circles from the tops of frozen ice cream cones as templates.

Olfa circle cutter. Perfect circle cutter up to 8 1/2 inch circles.