Heat a Dutch oven over medium heat, and sauté the bacon until browned and crisp (about 8 minutes). Transfer to a paper towel lined plate reserving 1 tablespoon of bacon fat, and set aside.
To the bacon grease add the celery, carrot, and onion, and cook while stirring occasionally for 3 minutes.
Add the garlic, and cook for 30 seconds.
Sprinkle the vegetables with flour, and sauté for another minute.
Add the chicken or vegetable stock/broth, clam juice, bay leaf, thyme, salt, and pepper. Cook for 1-2 minutes while whisking constantly until it thickens. Bring to a boil, then reduce the heat to bring to a simmer.
Add the potatoes, and then stir in the milk.
Cover with the lid and cook for about 15 minutes or until the potatoes are fork-tender.
Remove the bay leaf.
Stir in the cream and the clams, and cook for 2 more minutes (simmer, do not boil).
Adjust seasonings to your preference, and adjust the consistency if needed (add more milk if the chowder is too thick, or thicken with a cornstarch slurry if it’s too thin. Add 1-2 tablespoons of cornstarch mixed with the chowder broth and cook for 1-2 minutes to let it thicken).
Garnish with bacon and fresh parsley leaves, and serve immediately with toasted bread and oyster crackers.