Saucy Grilled Ham & Cheese "Croque Madame" with Pickled Veggies

Directions

1. Whisk together vinegar, sugar, and salt in a bowl until the sugar dissolves; this flavors the quick pickle.
2. Thinly slice the carrot and shallot, then add them to the pickling liquid to marinate briefly.
3. Slice the cucumber thinly and fold it into the bowl with the other vegetables, letting them mingle in the marinade.
4. Slice the mozzarella into thin pieces for even melting.
5. Spread Dijon on each bread slice; layer mozzarella on half the slices, top with prosciutto, then cap with the remaining bread slices.
6. Warm a nonstick skillet over medium heat, then melt a little butter and cook the sandwiches until the bottoms turn golden brown, about 2–4 minutes.
7. Flip the sandwiches, cook until the other side is golden and the cheese has melted, then transfer to a cutting board.
8. Wipe the skillet, melt more butter, and crack in the eggs; cook until the whites set and the yolks remain runny, about 3 minutes.
9. Meanwhile, heat a small saucepan over medium-high heat; add butter, whisk in flour with salt and pepper, cook briefly, then whisk in milk until the sauce thickens, about 2 minutes.
10. Slice the sandwiches, drizzle with the sauce, and top with an egg; serve with the pickled vegetables on the side.