Heat 2 tablespoons of olive oil in a medium skillet over medium heat.

Add the mushrooms and cook, stirring, until softened.

In a medium saucepan, heat the remaining 4 tablespoons of olive oil and butter over medium heat.

Add the flour and cook, stirring constantly, for about 3 minutes until the sauce is thickened and smooth.

Gradually stir in the milk, cider, and broth.

Cook, stirring occasionally, until the mixture is blended.

Season with salt and pepper.

Add the cooked mushrooms and cook, stirring occasionally, until the mushrooms soften and release their juices, incorporating them into the sauce.

Spread the mixture out on a dinner plate and let it cool, then refrigerate until firm enough to handle.

Shape the mushroom mixture into 1-inch balls.

Place the eggs and bread crumbs in separate shallow bowls.

Heat at least 1 inch of oil in a large skillet over medium-high heat (or use a deep fryer set at 365°F) until it turns a cube of bread light brown in 60 seconds.

Dip each croquette in the beaten egg, dredge in the bread crumbs, and place in the hot oil.

Cook in batches if needed to avoid overcrowding, turning the croquettes several times until golden.

Transfer the croquettes with a slotted spoon to paper towels and let them drain.

Serve hot.