HEAT olive oil over medium heat in a large fry pan. FRY the eggs according to preference. REMOVE from pan, PLACE on a sheet pan and SET in warm oven.
ADD onions to fry pan and SAUTÉ until transparent, about 5 minutes. ADD ground beef and break up with a large spoon. COOK for 7 minutes or until fully cooked. DRAIN fat from beef and return to pan.
WHISK together ketchup, water, vinegar, sugar, salt, pepper and paprika in a large bowl. ADD ketchup mixture to cooked beef and COVER. SIMMER over low heat for about 20 minutes, STIRRING occasionally. REMOVE from heat.
ADD 1/4 cup of sloppy joe mixture to each bun. REMOVE eggs from oven and PLACE on top of each one. COVER with top of bun. SERVE immediately.
For a deluxe version, add slices of cheese and cooked bacon on top of the fried eggs.
This recipe is an excellent source of protein, folate and choline, and a good source of vitamin A.