In a small bowl, combine the chives, tarragon, garlic, minced spinach, anchovy paste, if using, and 3 tablespoons of the mayonnaise until well incorporated.
In a separate small bowl, combine the butter with the remaining 4 tablespoons of mayonnaise. Spread one side of each slice of bread with about 1 tablespoon of the butter mixture.
Heat a large skillet over medium-low heat. Place a slice of bread, butter side down, in the pan and top with ¼ cup of the mozzarella. Add ¼ of the avocado slices, followed by ¼ of the spinach leaves, ¼ of the shaved asparagus, and ¼ of the microgreens, if using. Top with another ¼ cup mozzarella.
Spread 2 tablespoons of the green goddess mayonnaise onto the non-buttered side of one of the slices of bread. Place on top of the mozzarella, green goddess side down. Repeat with the remaining ingredients for as many sandwiches that will fit in the skillet.
Increase the heat to medium. Cover the pan and cook, flipping halfway through, until the cheese is melted and the sandwich is golden on both sides, about 12-14 minutes. Repeat with the remaining ingredients.