In shallow dish, whisk together flour, Cajun seasoning, salt and thyme. In bowl, whisk eggs with 2 tbsp (25 mL) water. In separate shallow dish, combine bread crumbs with Parmesan cheese.
Four at a time, toss drumettes in flour mixture to coat, shaking excess back into dish. Coat in egg mixture, letting excess drip back into dish. Roll in bread-crumb mixture, patting to adhere. Place on greased or parchment paper-lined rimmed baking sheets.
Bake in top and bottom thirds of 400°F (200°C) oven, turning drumettes once and switching and rotating pans halfway through, until juices run clear when chicken is pierced, about 40 minutes. (Make-ahead: Let cool for 30 minutes; refrigerate, uncovered, until cold. Cover and refrigerate for up to 1 day. Reheat in 400°F/200°C oven, about 10 minutes.) Serve with Blue Cheese Dip.