Bring a large pot of lightly salted water to a rolling boil. Cook the Swiss chard stems in boiling water until tender, 10 to 15 minutes; drain and rinse with cold water. Set aside.
Bring a separate large pot of lightly salted water to a rolling boil. Cook the vermicelli pasta in boiling water until tender yet firm to the bite, 4 to 5 minutes; drain.
Meanwhile, melt the butter with the olive oil in a large skillet over medium heat; stir in the onion and garlic scapes. Cook and stir until the onion is soft and translucent, about 5 minutes. Increase heat to medium-high, stir in the chard stalks, and cook until the onion has browned, about 5 minutes more. Season with salt and pepper; stir in the cooked pasta to serve.