In a medium saucepan over medium heat, melt 4 tablespoons of butter until foaming. Add the flour all at once and whisk vigorously for 2-3 minutes to create a roux, cooking until it turns light golden and smells nutty.

Gradually add the milk in small increments, whisking constantly to prevent lumps from forming. Continue whisking until the mixture is completely smooth and begins to thicken, about 5-7 minutes.

Reduce heat to low and stir in the shredded Parmesan cheese and half of the Swiss cheese. Season with salt, black pepper, and a generous pinch of nutmeg. Stir until the cheese is fully melted and the mixture is thick enough to pull away from the sides of the pan.

Remove from heat and transfer the mixture to a shallow dish greased with butter. Spread evenly, cover with plastic wrap touching the surface, and refrigerate for at least 3 hours or overnight until completely firm.

Once chilled, divide the mixture into 16 equal portions. With lightly oiled hands, shape each portion into a cylinder about 3 inches long and 1 inch in diameter. Place on a parchment-lined tray.

Set up a breading station with three shallow dishes: one with flour, one with beaten eggs, and one with breadcrumbs. Coat each croquette first in flour, shaking off excess, then dip in egg, and finally roll in breadcrumbs until completely covered. For extra crispiness, repeat the egg and breadcrumb coating.

Heat vegetable oil in a deep pot or dutch oven to 350°F (175°C). The oil should be deep enough to fully submerge the croquettes.

Carefully lower 3-4 croquettes at a time into the hot oil using a slotted spoon. Fry for 3-4 minutes, turning occasionally, until golden brown and crispy on all sides.

Remove with a slotted spoon and drain on a wire rack set over paper towels. Season immediately with a pinch of salt while still hot.

Serve the croquettes hot with lemon wedges and a side of Belgian mayonnaise or mustard for dipping.