Bring barley and 2 cups broth to a boil. Reduce heat to a simmer; cook, covered, until tender and most of the liquid has been absorbed, 40 to 50 minutes or whatever the package instructions suggest.
Drain and set aside. Meanwhile, heat the oil in a large non-stick skillet over medium-high heat. Add the onion and sauté stirring often until translucent, about 3 minutes. Add squash and sauté for 5 minutes. Mix in garlic and wine and cook until evaporated, about 1 minute. Add remaining broth, sage and thyme. Cover; bring to a boil. Reduce the heat and simmer, covered, 20 to 25 minutes or until squash is just tender.
Stir in barley and corn. Add spinach, lemon zest, and Parmesan. Season with salt and pepper. Remove from heat. Serve with additional Parmesan if you like.