Heat braiser or large pot over medium heat and add 2 tablespoons olive oil. Add mushrooms and rosemary and sauté, about 4 minutes. Scoop the mushrooms out of the braiser and set aside in a bowl.
In a pot, warm broth and pumpkin puree, stirring until well combined. Turn heat off.
While the broth is warming, heat the braiser over medium-low heat. Add remaining tablespoon of olive oil, then add onions and celery and sauté.
Once onion and celery are mostly cooked through, add garlic, ghee, and Arborio rice. Toss to coat the rice in the olive oil and ghee.
Add white wine, salt, and black pepper to the rice and stir to combine.
Add a ½ cup of the pumpkin broth to the rice, stirring constantly until the broth is absorbed. Repeat adding ½ cups of the broth at a time until rice is tender. This should take about 15-20 minutes. Turn heat off.
Add sautéed mushrooms and the parmesan to the rice and mix well. Serve immediately.