Whisk together eggs, flour, milk, olive oil, salt and black pepper until smooth batter forms. Let rest for 10 minutes.
Heat butter in a non-stick pan over medium-high heat. Pour a thin layer of batter into the pan, tilting to spread evenly.
Cook crepe for 2 minutes until light golden, then flip and cook another 1-2 minutes. Transfer to a plate. Repeat for remaining batter.
Slice tomatoes thinly. Tear mozzarella into pieces.
Reduce balsamic vinegar in a small saucepan over low heat until it thickens into a glaze, about 5 minutes.
Place a crepe on a plate. Layer with prosciutto ham, mozzarella cheese, tomato slices, and fresh basil leaves.
Top with fresh arugula and drizzle with balsamic glaze. Serve immediately while warm.
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