Heat 1 tbsp. oil in a large pot over medium heat. Add onion and 1/2 tsp. salt and cook until softened, stirring occasionally, about 8 minutes.
Roughly chop carrots and add to pot. Stir in broth and bring to a boil. Reduce to a simmer, cover (leave lid slightly ajar), and cook until carrots are very tender, about 20 minutes.
Using an immersion blender, blend soup until pureed. Alternate method: ladle soup into a blender and puree.
Add spinach, garlic, pine nuts, and nutritional yeast to food processor. Pulse until blended. Scrape down sides of bowl with a spatula. With food processor on, stream 2-3 Tbsp. olive oil into bowl until pesto mixture is to desired consistency.
Serve carrot soup with a dollop of fresh pesto.