Carefully transfer to a blender. Add fresh cilantro leaves and puree until smooth. Season with salt and fresh ground black pepper. Beat eggs with milk. Scramble eggs in a small amount of oil with diced ham.
Heat tortillas on a flat griddle.
In the center of each tortilla portion scrambled eggs and ham. Top with salsa verde, diced tomato and shredded cheeses. Roll and fold in ends to form burritos. Cut each burrito in half and place on warm serving plates. Drizzle with additional salsa verde and garnish with cilantro sprigs.