Combine spinach, goat cheese, avocados and garlic in the bowl of a food processor; season with salt and pepper, to taste. With the motor running, add olive oil in a slow stream until emulsified. Stir in mozzarella.
With a butter knife, spread butter on one side of each slice of bread. Place 4 slices, buttered side down, on work surface. Divide avocado mixture evenly on all slices. Top with remaining slices, buttered side up.
Heat a large skillet over low heat. Add sandwiches and cover; cook, turning once with a spatula, until golden brown and the cheese has just melted, about 2-3 minutes per side.