In a small saucepan over medium heat, melt the butter. Add the flour to the pan and whisk to combine. Reduce heat to low and cook until the mixture smells lightly nutty.
Add milk to the pan and whisk to combine. Cook for 2-3 minutes, whisking consistently, until mixture becomes nappe.
Remove from heat and set aside momentarily.
In a medium mixing bowl combine the corned beef hash, salt, pepper, and cayenne powder. Pour prepared bechamel over the top of the mixture. Mix until thoroughly combined. Cover with plastic wrap and place in the refrigerator to chill for 1 hour.
Line a sheet tray with parchment paper. Take mixture and divide into 18 – 2 Tbsps. portions. Round out each portion and line up on the parchment lined sheet tray.
Place the flour, eggs, and breadcrumbs in separate bowls. Working with one at a time, roll each ball in flour, followed by a dip in the beaten egg and then a roll in the breadcrumbs. Loosely roll balls in hand to ensure the coating is secure.
Slide each croquette onto a bamboo skewer and return to the sheet tray.
Place croquettes in the fridge for 30 minutes prior to frying.
Preheat fryer to 325F. Deep fry the croquettes until golden brown and crispy, ~2 minutes.
Transfer croquettes to a cooling-rack-lined sheet tray. Sprinkle with salt. Hot hold while the cheddar cheese dipping sauce is prepared.
For the cheddar cheese dipping sauce, combine the basic cheddar cheese sauce and whole milk in a pan. Heat over medium-low heat and whisk to combine. Once warm, transfer to a serving dish.
Place croquettes on a serving dish with the warm cheddar cheese dipping sauce. Serve immediately.