Chop the onion, potato, carrots, and celery into pieces that are about 1/2-inch or smaller. Do your best to cut everything roughly the same size, this will keep cooking time across the ingredients somewhat uniform. You should end up with about 2 1/2 cups of chopped celery.
In a large thick-bottomed soup pot over medium high heat combine the olive oil, onion, potato, carrots, celery and a few big pinches of salt. Saute for about ten minutes or until the onions and celery soften a touch and expel some water.
Stir in the garlic and add the stock. Bring to a simmer and let cook for another 10 minutes or until the celery, carrot and potatoes are just cooked through - resist the urge to over cook them into mush. Stir in the rice a few minutes before the potatoes and carrot are cooked though.
Process ingredients for the celery pesto in food processor for about 30 seconds. Ladle soup into bowls and top with the celery pesto. Enjoy!