Skipping the boiling water and cooking frozen pierogis in a hot skillet is an all-time favorite shortcut. Not only does it lessen the cooking time in this recipe, it gives the pierogi shell more time to get really crispy, which is a great contrast to the creamy filling.
Heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Add the sausage and cook, breaking it up into small clumps with a spatula or wooden spoon, until browned and cooked through, about 8 minutes. Remove to a large plate, leaving the drippings in the pan. Add the pierogi to the pan and cook, turning, until browned and heated through, about 5 minutes. Remove to the plate with the sausage.
Add the onion, oregano and a big pinch each of salt and pepper to the skillet. Cook, stirring, until the onion is softened and just beginning to brown, about 3 minutes. Push to one side of the skillet; add 1 more tablespoon olive oil and the broccoli to the open side. Season the broccoli with salt and let cook, undisturbed, until it starts to brown in spots, 2 to 3 minutes. Stir the broccoli into the onion and cook until the broccoli is bright green with crisp-tender stems, 5 minutes.
Add the bell pepper to the skillet and cook until just softened, 2 to 3 minutes. Return the sausage and pierogi to the skillet and add the garlic and remaining 1 tablespoon olive oil. Cook, tossing, until well combined. Season with salt and pepper and stir in 2 tablespoons Parmesan.
Divide the pierogi and vegetables among plates or shallow bowls. Sprinkle with the remaining 2 tablespoons Parmesan.