How to Make It
In a large skillet, melt 2 tablespoons of the butter. Add the leeks and shiitake, season with salt and pepper and cook over high heat, stirring frequently, until the leeks and mushrooms are softened and beginning to brown, about 7 minutes. Add the thyme and oregano to the vegetables and transfer to a medium bowl.
In a small saucepan, melt the remaining 3 tablespoons of butter. Whisk in the 3 tablespoons of flour and cook over high heat until bubbling, about 1 minute. Add the milk and cook, whisking, until very thick and bubbling, about 3 minutes. Scrape the mixture into the bowl. Add the Gruyère and Parmigiano-Reggiano, season with salt and pepper and stir until the mixture is evenly combined.
Lay an 18-inch-long piece of plastic wrap on a work surface. Spoon the croquette mixture onto the plastic in a 12-inch strip. Roll up the plastic, pressing the croquette mixture into a 14-inch log, and twist the ends. Freeze the croquette log until very firm, about 2 hours.
Fill 3 shallow bowls with flour, the beaten eggs and the panko and line a large baking sheet with wax paper. Unwrap the log and cut the croquette mixture into 12 pieces. Using floured hands, pat each piece into a 2-inch round patty, about 3/4 inch thick. Dip each patty in the flour, then dip in the eggs and coat with panko, pressing to help it adhere. Set the croquettes on the baking sheet and freeze for 15 minutes.
In a large skillet, heat 1/2 inch of oil to 375°. Add all of the croquettes and fry over high heat, turning once or twice, until they are golden and crisp, about 5 minutes. Drain on paper towels and serve hot.
A green salad.