Heat the oil in a saucepan over medium heat. Add the onion, ½ teaspoon salt, and ¼ teaspoon pepper and cook, stirring occasionally, until tender, 6 to 8 minutes.
Add the barley, tomatoes, wine, and 2 ½ cups water and bring to a boil.
Reduce heat and simmer, stirring occasionally, until the barley is tender, 12 to 15 minutes.
Add the mustard greens, Taleggio, and ¼ cup of the Parmesan and cook, stirring occasionally, until the greens are tender, 3 to 4 minutes.
Divide among bowls and sprinkle with the remaining ¼ cup of Parmesan.