Preheat the oven to 425 degrees F and line a rimmed baking sheet with foil. Pulse the bread with half of the mushrooms in a food processor until fine; transfer to a large bowl. Add the beef, egg, 1 tablespoon parsley, the paprika, Worcestershire sauce, 1/2 teaspoon salt and a few grinds of pepper. Mix with your hands until just combined. Form into four 3-by-4-inch mini meatloaves; transfer to the prepared baking sheet. Bake until browned and cooked through, about 15 minutes.
Meanwhile, bring a medium saucepan of salted water to a boil. Add the orzo and cook as the label directs. Reserve 3/4 cup cooking water, then drain the orzo.
Slice the remaining mushrooms. Heat 1 tablespoon olive oil in a large nonstick skillet over high heat. Add the mushrooms and cook, undisturbed, until they just start browning, 2 to 3 minutes. Stir in the sage and season with salt and pepper; cook until the mushrooms are tender, 1 to 2 more minutes. Stir in the orzo.
Remove the skillet from the heat and stir in the remaining 3 tablespoons parsley and 1 teaspoon olive oil. Gradually add the reserved cooking water as needed to loosen; season with salt and pepper. Drizzle the meatloaves with the yogurt and sprinkle with paprika. Serve with the orzo.