Place the rice in a large microwave-safe bowl and add 1 cup water. Cover with plastic wrap, pierce with a knife a few times to vent, then microwave until the water is almost absorbed, 10 minutes. Uncover and fluff the rice with a fork; let cool completely.
Lightly beat 1 egg in a small bowl. Add to the bowl with the rice along with the lamb, beef, provolone, onion, half each of the mint and parsley, the marjoram, garlic, allspice, 1 teaspoon salt and 1/2 teaspoon pepper. Mix with your hands until just combined (do not overmix); cover and refrigerate 30 minutes.
Dampen your hands and form the meat into 18 balls (about 1 1/2 inches each). Bring the chicken broth to a boil in a large pot or Dutch oven. Add the meatballs, cover and cook over medium heat until tender, about 30 minutes, turning halfway through.
Whisk the remaining 2 eggs and the lemon juice in a small bowl until frothy. Remove 1 cup broth from the pot and gradually whisk it into the egg mixture. Push the meatballs to one side of the pot; reduce the heat to low and pour the egg mixture into the other side. Cook, stirring gently, until the sauce thickens slightly, about 4 minutes. Remove from the heat and stir in the remaining mint and parsley; season with salt and pepper. Serve the meatballs and sauce in shallow bowls. Top with arugula, drizzle with olive oil and season with salt and pepper.