Melt 4 tablespoons (1/2 stick) of the butter in a large heavy-bottomed stockpot. Add the onions and cook over medium-low heat for 10 minutes, or until translucent. Add the celery, carrots, potatoes, thyme, salt, and pepper and saute for 10 more minutes. Add the clam juice, bring to a boil, and simmer, uncovered, until the vegetables are tender, about 20 minutes.
In a small pot, melt the remaining 8 tablespoons of butter and whisk in the flour. Cook over very low heat for 3 minutes, stirring constantly. Whisk in a cup of the hot broth and then pour this mixture back into the cooked vegetables. Simmer for a few minutes until the broth is thickened.
Add the milk and clams and heat gently for a few minutes to cook the clams. Taste for salt and pepper. Serve hot.
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