Heat 2 tablespoons oil in a large skillet over medium-high heat. Add eggplant and next 4 ingredients; cook 7 minutes or until vegetables are tender, stirring occasionally. Add crawfish; cook 3 minutes. Remove from heat, and cool.
Pour oil to depth of 3 inches in a Dutch oven; heat to 360°.
Whisk together eggs and next 3 ingredients; whisk in flour just until moistened. Fold in eggplant mixture. Drop batter by heaping tablespoonfuls into hot oil. Fry beignets, a few at a time, 3 minutes or until golden, turning once. Drain on paper towels. Serve immediately with Remoulade Sauce.