Heat oil in skillet: Heat 1 tablespoon of extra virgin olive oil in a large skillet over medium heat. Add onions and season with salt and pepper. Cook, stirring occasionally, are almost softened, 3-4 minutes.
Add in the mushrooms: Stir the mushrooms into the onions and they soften, about 4-5 minutes. Remove onions and mushrooms and add them to a small bowl (that way the mushrooms don't overcook and get too mushy).
Cook the ground turkey: In the same skillet on medium heat, add another tablespoon of extra virgin olive oil to the pan, and add in the ground turkey. Use a wooden spoon to break it up into small pieces (the ground turkey will take about 14-16 minutes in total to cook, but cook it until it’s almost ready, about 12 minutes), When it has just a little bit of pink color, go on to the next step.
Add broth: Add in the broth and a splash of white wine (you could leave that out if you don't drink or don't have any on hand. The alcohol evaporates while it's cooking). Next, add in the fresh rosemary.
Finish cooking ground turkey: Cook that all together about 5 minutes. The turkey will finish cooking with the wine and broth.
Boil pasta: While the ground turkey is cooking, cook up the pasta. Fill a large pot with water and add salt (to cook the pasta)and bring to a boil. When it boils, add in your pasta and cook to package directions just until one minute before it reaches al dente.
Stir in pumpkin: Add in the pumpkin puree, salt, freshly ground pepper, a dash of cinnamon, ground nutmeg (I grated fresh nutmeg right on top -about a 1/2 teaspoon). Stir it all together and cook it about 5-7 minutes.
Add back mushrooms + onions: When the sauce is cooked up, add back in the mushrooms and onions. Stir it all together and let it cook together for about 1-2 minutes.
Strain the pasta: When the pasta is one minute before al dente, reserve 1 cup of the pasta water. Strain the pasta in a colander and add pasta immediately to the skillet with the turkey/pumpkin sauce.
Check sauce consistency: If it's too thick, you can loosen it up with a bit of the pasta water. Stir it all together to combine on medium-high heat (about 1 minute). Remove the rosemary sprig.
Serve pasta: Plate up and serve with some freshly ground black pepper (if you like)and rosemary leaves. Optional: spice it up with some hot pepper flakes and grated cheese (if you're not dairy-free).