Preheat the oven to 180°C/350°F. Line a baking sheet with parchment paper.
Remove the very inner part of the zucchini flower (the pistil) and the tiny pointed green parts on the outside, leaving the outer yellow-orange flower, including the green bottom, as intact as possible. Look carefully inside and out for any small bugs or debris and clean gently if needed.
Prepare the filling: Combine the mozzarella, anchovies, and parmesan in a food processor and blend until smooth. Transfer the filling to a small bowl. Carefully open the flowers and stuff each one with 1-2 tablespoons of the filling until full. Pinch each flower closed at the top and set aside.
Pour the beaten egg into a bowl and add the breadcrumbs to a separate bowl. Dip each flower first in the egg, letting any excess drip off, and then dip in the breadcrumbs, turning to coat the flower completely.
Arrange the zucchini flowers on the baking sheet and drizzle them with olive oil. Bake for about 20 minutes, or until the flowers are golden-brown and crispy on the outside. Serve immediately.