Prep all ingredients as listed.
Slice the Brie cheese and allow it to sit at room temperature until softened for easier spreading.
In a bowl, mix the softened butter, minced chives, minced parsley, Dijon mustard, and grated garlic until well combined. Season with salt and black pepper to taste.
In a separate bowl, toss the arugula with lemon juice and season lightly with salt and black pepper.
Spread the herb butter mixture inside the rolls.
Layer with deli ham, Brie cheese, and arugula, then top with the remaining half to form the sandwich. Cut in half if preferred.
Add lemon slices to the sparkling water, add ice if desired, and serve chilled.
Serve the jambon beurre alongside carrot sticks, hummus, and lemon sparkling water.