I recently had a delicious beet mimosa at Bar Tartine. I have no idea how they made theirs, but it inspired me to try one at home.
1 oz. fresh beet juice
1/2 oz. thyme-infused simple syrup
Run some beets through a juicer. Make a 1:1 simple syrup and throw some thyme in there while it cools. (See the Fall Classic. If you make enough, you can have Fall Classics by night and Beet Mimosas by day.) Pour the bubbles in a glass, then add the juice and syrup. Warning: When beet juice meets bubbly, the result is volcanic. I recommend making the drinks on a surface that won't stain.