Melt butter in a large pot over medium heat. Cook and stir onion, celery, thyme, salt, and pepper in hot butter until onion and celery start to soften, about 5 minutes. Stir potatoes, milk, broth, cream, and clam juice into onion mixture. Bring to a simmer; cook until potatoes are soft, about 10 minutes.
Pour broth mixture into a blender no more than half full. Cover and hold lid down; pulse a few times before leaving on to blend. Puree in batches until smooth. Return pureed soup to pot and stir in crabmeat. Cook until soup is hot and crab is heated through, about 5 minutes. Season with salt and pepper.