Heat 1 tablespoon butter and 1 tablespoon oil in a 12-inch saute pan over medium-high heat. When hot, add the shallots and cook, stirring, until tender, 1 to 2 minutes. Add the barley, 1 1/2 teaspoons salt and 1/2 teaspoon pepper and cook for 1 minute. Pour in the white wine and cook, stirring, until the wine is evaporated, 1 to 2 minutes. Transfer to the crock of a 6-quart slow cooker.
Add 2 tablespoons butter and the remaining 1 tablespoon oil to the saute pan over medium-high heat. Add the mushrooms and cook until lightly browned and wilted and the fat has been absorbed, 4 to 5 minutes.
Add the mushrooms, squash and chicken stock to the slow cooker. Stir gently to combine. Set the slow cooker to low and cook until the liquid is absorbed and the barley and squash are tender, 2 hours, 30 minutes to 3 hours, stirring occasionally to ensure all of the grains are evenly cooked. Remove the lid and stir in the parsley, sage, 1/2 cup parmigiano-reggiano and the remaining 2 tablespoons butter. Serve the risotto hot, garnished with the remaining 1/4 cup cheese and additional pepper, if desired.