Cook the tortellini according to the package directions. Drain.
Meanwhile, heat the oil in a large skillet over medium heat. Add the onion and cook until soft, about 6 minutes.
Increase heat to medium-high. Add the butter and cook until it turns golden brown, about 3 minutes.
Stir in the garlic, peas, ½ teaspoon salt, and ¼ teaspoon pepper and cook until heated through, 2 to 3 minutes.
Add the tortellini, tarragon, and half the Parmesan to the skillet and toss. Divide among bowls and sprinkle with the remaining Parmesan.