Melt butter in a skillet over medium heat. Saute onion until softened, about 3 minutes. Add beet stems; saute for 1 to 2 minutes. Add white wine and cook, stirring constantly, for 1 minute. Add rice and stir until it begins turning translucent, about 2 minutes.
Scrape rice and onion mixture into a slow cooker. Set slow cooker on High. Add beet greens and broth. Cover and cook until rice is tender but not soupy, 1 3/4 to 2 1/2 hours. Stir in Parmesan cheese.
