In a large, heavy-bottomed pot, melt butter over medium heat. Add diced onions and cook for 5 - 6 minutes until they begin to soften. Add diced celery and cook for an additional 2 - 3 minutes.
Add reserved corn cobs, lower the heat and cover. Simmer for 25 - 30 minutes on medium-low heat.
After 30 minutes, remove corn cobs carefully with kitchen tongs and discard.