Halve stripped corn cobs; put in a large saucepan with stock water. Simmer partly covered 20 minutes. Strain; discard cobs. Measure 4 cups stock, adding water if needed.
Meanwhile, melt butter in a soup pot over medium. Cook onion and celery 7 minutes until soft, not browned.
Add potatoes, measured stock, salt, pepper, and thyme. Gently simmer 12 to 15 minutes until potatoes are almost tender.
Add corn; simmer 5 minutes. Carefully puree 2 cups soup with plenty of corn and potato in a heat-safe container until smooth. Return to pot.
Add milk and cream; warm on low 3 to 5 minutes until steaming without a hard boil. Divide among 6 bowls, retaining kernels and potato cubes in each.