Bring a large pot of salted water to a boil. Cook the pierogi as the label directs. Reserve 3/4 cup cooking water, then drain.
Meanwhile, heat the olive oil in a large nonstick skillet over medium-high heat. Add the sausage and cook, breaking it up with a wooden spoon, until browned, 6 to 8 minutes. Reduce the heat to medium. Add the bell peppers, onion, garlic and red pepper flakes and cook, stirring, until the vegetables are tender, 7 to 9 minutes. Season with salt.
Add the pierogi to the skillet and toss. Add the butter and 1/4 cup of the reserved cooking water; cook over medium heat, stirring gently, until the butter melts, adding more cooking water as needed to loosen. Stir in the basil.