Place potatoes, onion, celery, 1/4 cup butter, bay leaves, and thyme in a large saucepan with water. Bring to a boil, reduce heat, and simmer until vegetables are tender, about 20 minutes.
Melt remaining 1/2 cup butter over medium-low heat. Whisk flour into melted butter for 2 minutes; slowly whisk half-and-half into flour mixture and season with Cajun seasoning, salt, and black pepper. Whisk continuously to prevent boiling until mixture is thickened. Stir in potato mixture and crawfish tails; cook, stirring constantly, until heated through, about 5 minutes.