Place vinegar, sugar, and salt in a medium bowl; whisk to combine the marinade.
½ cup rice vinegar
2 tsp brown sugar
1 tsp salt
Wash, dry, peel, trim, and thinly slice carrots. Peel and thinly slice shallots; add both to the bowl with the marinade.
2 medium carrots
2 shallots
Wash, dry, trim, and thinly slice cucumber; add to the bowl, toss to combine, and set the veggies aside to marinate.
1 English cucumber
Thinly slice mozzarella.
1 (8 oz) block mozzarella cheese
Place the bread slices on a flat surface and spread with Dijon. Evenly top half the slices with the mozzarella and prosciutto. Cover the sandwiches with remaining bread slices.
8 slices whole grain bread
4 tsp Dijon mustard
12 slices prosciutto
Preheat a large skillet over medium heat.
Once the skillet is hot, add butter and swirl to coat the bottom. Working in batches if necessary, place sandwiches in the skillet and cook until the bottoms are golden brown, 2-4 minutes.
2 tsp butter, unsalted
Remove sandwiches to a plate and add more butter to the skillet. Return sandwiches to the skillet, browned-side up; cook until golden brown on the other side and the cheese is melted, 2-4 minutes. Once done, transfer to a cutting board.
2 tsp butter, unsalted
Wipe the skillet clean with a paper towel; return to medium heat, and add more butter. Gently crack eggs into the skillet and cook until whites are set but yolks are still runny, about 3 minutes. Remove from heat and season with salt and pepper.
2 tsp butter, unsalted
4 eggs
⅛ tsp salt
⅛ tsp black pepper
Meanwhile, preheat a small saucepan over medium-high heat.
Once the saucepan is hot, add butter and swirl to coat the bottom. Add flour, salt, and pepper; cook, stirring constantly, until fragrant, about 1 minute.
2 tbsp butter, unsalted
2 tbsp all-purpose flour
¼ tsp salt
⅛ tsp black pepper
Slowly whisk milk into the saucepan; bring to a boil and cook until sauce is thickened, 2-3 minutes.
8 fl oz (1 cup) whole milk
Divide sandwiches between plates; drizzle with sauce and top with a fried egg. Serve with pickled veggies on the side and enjoy!