Add all pesto ingredients to a food processor and blend until smooth. Season with salt and pepper to taste and set aside.
Slice croissants in half and lightly drizzle with oil. Toast cut-side down in a skillet over medium heat until golden. Remove and set aside.
In a bowl, whisk eggs with milk. Heat butter or oil in the skillet and scramble eggs over medium heat until just set. Season lightly.
Spread pesto generously on the toasted croissants. Layer scrambled eggs, tomato slices, mozzarella, and arugula on the bottom half.
Drizzle lightly with balsamic vinegar glaze and top with the remaining croissant. Serve immediately
Notes
This sandwich pairs beautifully with fresh fruit, roasted potatoes, or a light green salad. Pesto can be made up to 3 days in advance and stored refrigerated.