Melt butter over medium heat in a large Dutch oven or soup pot. Add oil, chopped shallot, and onion, and cook until soft and transparent, 4-5 minutes.
Add chicken broth, corn cobs, and potato. Bring to a boil, then reduce to a simmer and cook for for 15 minutes until potato is very tender. Remove cobs, add corn and return to a simmer. Cook for another 2 minutes.
Allow soup to cool slightly, then purée in batches in a blender or food processor until smooth. Place a fine sieve over a medium-size bowl and strain, pressing on solids with the back of a spoon. Discard solids.
Return soup to pot, add salt and a generous grind of black pepper. Rewarm over medium heat. Serve in bowls garnished with fresh corn and drizzled with pesto oil.