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.
This is great with oyster crackers! It's also even better left over! Cayenne pepper or other red pepper seasonings can be substituted for the Cajun seasoning. Just be careful with children as this dish can be very spicy depending on the seasoning used!