Cut one end off each zucchini. Use a small scoop to scoop the flesh from the zucchinis, leaving shells. Discard the flesh.
Mix chopped round steak, rice, nutmeg, salt, and black pepper together in a bowl. Lightly stuff the steak mixture into the zucchinis, making sure to leave about 1 inch of space for the rice to expand; if you don't, the zucchini will burst which still tastes good but doesn't look as nice.
Place the zucchinis in a large pot; add tomato soup and stewed tomatoes. Place a plate on top of the zucchinis to prevent them from bouncing around once the sauce is boiling.
Bring the sauce to a boil, reduce heat to medium-low, and cook at a simmer until the rice is fully cooked, 30 to 45 minutes.