Prepare the pesto by combining all ingredients except olive oil in a food processor. As the pesto is being processed, slowly add olive oil until emulsified and you’ve reached the desired consistency. Set aside.
In a large bowl, whisk eggs. Heat a pan on medium with 2 tablespoons of butter. Add eggs, season with salt and pepper and scramble.
Assemble the grilled cheese by buttering the outside of the bread, spreading pesto on the inside, and stuffing with tomatoes, spinach, egg, and the cheeses.
In a pan, toast the grilled cheese on each side until golden brown.