Chill the oranges for an hour. Peel and slice the oranges into 1/4 to 1/2 inch slices. I also find it easier to slice the chilled blood oranges with the peel on, then trim the peel off, being sure to get all the white pith.
Cut and peel the red onion and trim the ends off. Cut it in half, slice one half thinly, using about a quarter of the entire onion. Save remaining onion for other use.
Divide and arrange oranges slices on each plate, divide and sprinkle onion over the oranges and season with salt, pepper and drizzle olive oil over top. You can also drizzle some balsamic syrup on top as well to punch up the flavor (optional).
I often times will also sprinkle on ground sumac, which is a Middle Eastern spice with a lemony taste.