Cut the corn off the cob and use the back of your knife to scrape down to remove all the 'milk' and small bits of corn leftover in the cob. Set the corn aside, but don’t throw the cobs away because you’ll use them to flavor the broth later.
Heat oil and butter over medium heat in a large soup pot or Dutch oven.
Add the onion and celery and cook for 2 minutes to soften.
Stir in the garlic, 1 teaspoon salt, pepper, thyme, and cayenne pepper. Cook for an additional minute to combine the flavors.
Pour in the vegetable broth then add the corn cobbs to the pot. (You may have to snap the cobbs in half to make them fit.) Simmer for 10 to 15 minutes.
Use tongs to remove the corn cobbs and discard.
Add the potatoes and simmer for 10 minutes or until they are ¾ of the way cooked.
Stir in the corn kernels, milk, and remaining salt. Simmer on low until the potatoes are fully cooked and the corn is tender, about 5 minutes.
Turn off the heat. Taste the soup and add more salt and pepper as needed.
Stir in the chives then serve.
