Melt the butter in a large skillet over medium heat, and cook the celery and shallots until shallots are tender.
Pour half-and-half into a large pot over medium-high heat. Mix in the butter, celery, and shallot mixture. Stir continuously. When the mixture is almost boiling, pour the oysters and their liquid into the pot. Season with salt, pepper, and cayenne pepper. Stir continuously until the oysters curl at the ends. When the oysters curl the stew is finished cooking; turn off the heat and serve.