Heat oil over medium heat. Add onions & celery, stirring occasionally. Cook until soft & fragrant, about 5 minutes.
Add diced potatoes to the pot. Pour in chicken stock just until all the vegetables are covered. Cover & cook over medium heat for 15-20 minutes, or until the vegetables can be easily pierced with a fork.
Remove the pot from the heat. Now either mash with a potato masher or use an immersion blender to blend up the soup. (I use the masher if the potatoes were peeled and the immersion blender if the potatoes weren't peeled.)
Return to low heat. Pour in the half & half, shredded cheese, & butter. Stir until everything is mixed well, and the cheese & butter are melted. Add seasonings. Serve with garlic bread & listen to the rave reviews from your dinner guests & family.