Melt butter in a Dutch oven over medium-high heat; add onion and celery, and sauté until tender. Whisk in flour; cook 1 minute.
Gradually whisk in clam juice, broth, and milk. Stir in potatoes and carrot. Bring to a boil; cover, reduce heat, and simmer 20 minutes or until potatoes are tender.
Stir in salmon, clams, cheese, and half-and-half. Cook, stirring constantly, 1 to 2 minutes or until cheese melts. Stir in salt and pepper. Ladle soup into individual serving bowls.