In a Dutch oven, melt butter and olive oil over medium low heat. Add celery, onion, white portion of green onions and salt. Cook partially covered for 8 minutes, stirring occasionally.
Uncover the vegetables and add garlic. Continue cooking uncovered stirring occasionally for 8 minutes.
In a separate bowl, combine milk, thyme and cornstarch. Stir until well combine and the mixture is smooth. Let stand for a few minutes.
Once the vegetables are tender and soft, stir the milk mixture vigorously with a fork to make sure there are no clumps. Pour into the vegetable mixture and stir.
Bring the soup to a boil and then reduce heat to a simmer. Add the chicken stock to thin. Continue to simmer for 25-30 minutes.
Salt and pepper to taste. Garnish with green onions and serve.