Heat the oven to 400°F. Toss the squash with 2 tablespoons olive oil and season with salt and pepper. Place on a parchment-lined baking sheet and roast for 25–30 minutes until the edges are browned and crisp. Remove from oven and immediately sprinkle 1 tablespoon sage over the still hot squash.
Meanwhile, bring 4 cups of water to a boil over medium heat. Stir in your farro and cook for 15 minutes. Drain, reserving 1 cup of the cooking liquid, and set aside.
In a 4 quart pan, heat the remaining 1 tablespoon olive oil over medium heat. Add in the chopped shiitakes and sauté, about 3 minutes, remove from pan and set aside.
Melt the butter in the same pan over medium heat. Add in the shallots, garlic and thyme. Let cook until the shallots become slightly caramelized, about 3 minutes. Add the farro and stir, coating each kernel with butter. Add in the white wine and let simmer off.
Turn the heat down to low and begin alternately adding in the chicken stock and the farro cooking liquid, a 1⁄4 cup at a time, stirring continuously with each addition. Do not add more liquid until the previous 1⁄4 cup has been completely absorbed by the farro. This process should take about 25–30 minutes. After the last addition, stir until about 2⁄3 of the liquid has been absorbed, and add in the mushrooms and roasted squash.
Fold in the cheese and stir until just melted. Fold in the crème fraiche and remaining chopped sage for garnish, season with salt and pepper.