I’ve just finished my second little round Chicken and Egg Container and although I can improve on the design further I think I can write a tutorial incorporating those improvements so here goes, I hope it works out!
I don’t have a decent compass so I used plates for template but if you are more mathematically minded than I am the second circle for the lid should have a diameter that is a centimetre larger than your base. You can make the container any size measuring the circumference of your circles to make the sides the right size.
The measurements for my small container are:
Base diameter 5 and three quarter inches.
Lid diameter 6 inches.
Base sides 4 inches x 17 inches
Lid sides 1 and three quarter inches x 19 inches
1 large circle in the outer fabric, inner fabric and wadding for the lid. (3 circles) I trimmed the lining fabric an eighth of an inch smaller.
1 small circle in the outer fabric, inner fabric and wadding for the base. (3 circles)
 4″ x 17″ rectangles in outer fabric, lining fabric and wadding (base sides)
 1 and three quarter x 19″ strip in outer fabric, lining fabric and wadding
 1 and a quarter x 18″  1 and a quarter x 20″ strips binding fabric
 5″ x 2 and a half inch piece for the handle.
Pin the wadding to the outer pieces and sew in place as close to the edges as possible.
Quilt each piece before assembling. I decided on a simple square quilting and to give my new template a go. I found it really useful and of course I used a trusty Frixion erasable pen for marking. The markings come off so easily with just a touch of the iron.
Next sew together the 4 circles and their corresponding side pieces using quarter inch seams throughout. After sewing snips little nicks into the outer edge of the circle to allow some stretch and prevent puckering.
Sew the end seams of each of the side pieces to complete the shapes then drop the inner shape into the outer ones. The inner shape fits better if you have made the circle one eighth of an inch smaller (side piece will be slightly too long and will need trimming, I did this after sewing together to get the size right.
Pin around the tops of the base and lid, the inner shape will need to be stretched very slightly if you have made it smaller. Place the edge to be stretched closest to the plate and ease gently as you sew the layers together close to the edge.
Fold down and press a quarter inch hem along one edge of the binding strip. Place the unhemmed edge right side to right side around the rim of the container, folding a hem at the start of the strip. Sew around the rim using a quarter inch seam. Turn the binding over the top of the container and pin in place. There will be enough turn-over to allow you to stitch in place by stitching the ditch on the outside of the container like this:Alternatively you can hand sew the second binding edge in place. Repeat the binding process for the lid.
Make a handle by pressing a quarter inch hem around the fabric piece, you can trim the corners to make them sit tidily. Fold in half and press then sew around close to the edge. Pin and sew into place on the lid.It was a happy accident that I found that my two containers nested snuggly together. Handy if I use them in the van as Jeff suggested.I went ahead and made a lidless version. I just made the base 2 inches deeper and then turned over the top. I was careful to decide how I wanted the fabric to lie for this (so my chickens were upside down) and I also made sure the binding would look right by sewing it to the inside first and folding over to the outside. Here is the process in pictures.