Heat oven to 450°F. Line a rimmed baking sheet with aluminum foil. Toss squash with a teaspoon of oil, a sprinkle of salt, and a few grinds of pepper. Spread squash on baking sheet and roast for about 30 minutes, until tender. Set aside.
Add remaining 2 teaspoons of oil to a large saucepan and heat to medium-low. Add onion and cook until soft and translucent, about 7 minutes. Add garlic and cook until fragrant. Stir in pearl barley and increase heat to medium. Stir barley until lightly toasted and fragrant. Add wine and continue to cook, stirring often, until wine has been absorbed.
Stir in half of the chicken broth and half of the roasted squash. Bring to a simmer and cook, stirring occasionally, until liquid has been absorbed. Stir in another cup of broth and continue to cook, stirring occasionally, until liquid has been absorbed. Taste barley for texture (it should be cooked through but al dente at the center). Add remaining broth and water and continue to cook as needed.
Once barley is done, remove from heat and stir in the remaining squash, cheese, butter, herbs, and kale. Cover and let sit for 5 minutes. Season with salt and pepper to taste and serve.