Bring a large pot of lightly salted water to a rolling boil; stir in tortellini and return to a boil. Cook uncovered, stirring occasionally, until the tortellini float to the top and the filling is hot according to package directions. Drain and set aside.

Whisk oil, lemon juice, vinegar, parsley, oregano, and salt together in a large salad bowl; add tortellini and toss to coat. Cover bowl with plastic wrap; refrigerate, at least 2 hours.

Place eggs in a saucepan and cover with water. Bring to a boil, remove from heat, and let eggs stand in hot water for 10 to 12 minutes. Remove eggs from hot water, cool under cold running water, peel, and quarter.