Heat a large skillet over medium-high heat. Cook and stir ground beef in the hot skillet until browned and crumbly, 5 to 7 minutes. Drain and discard grease.

Place 6 cups water and brown rice into a stockpot. Add browned beef, bones, and rosemary; bring to a boil. Reduce heat to medium-low or low and let simmer, stirring occasionally to prevent burning, until rice is mushy, about 45 minutes. Add water if necessary to keep mixture a little thicker than a porridge-like consistency.

Meanwhile, place carrots, spinach, and eggs with their shells into an electric blender. Blend until completely pureed and no eggshells are intact.

Mix pumpkin and blended egg mixture in a bowl until combined. Remove meat mixture from heat and pour egg mixture into the pot to cook the eggs in hot contents, about 5 minutes.