Add 1 Tbsp olive oil to a large pot and set over medium heat. Add ½ cup diced white onion and 2 cloves garlic, cooking until soft and fragrant, about 5 minutes.
Add 1 ½ cups vegetable broth, 3 cups diced butternut squash, 1 tsp salt, and ¼ tsp pepper. Cover and let simmer until butternut is tender, 7 to 10 minutes.
Meanwhile, boil 1 lb uncooked pasta according to instructions on package, cooking until al dente.
Using a handheld immersion blender (or working carefully with a countertop blender), puree the butternut mixture until smooth.
Mix the butternut puree into the pasta. Add 1 cup shredded sharp cheddar cheese and stir to evenly combine, then add ½ cup plain yogurt and continue stirring until evenly mixed. Taste and add more salt and pepper as needed. Optionally stir in dijon mustard.
