In a small bowl, blend the butter with the Parmesan and season with salt and pepper. Spread 1 side of each bread slice with the Parmesan butter. In another small bowl, blend the mayonnaise with the mustard. Halve and pit the avocados and thinly slice them lengthwise. Sprinkle the slices with the lime juice.
Put 2 to 3 slices of bread, buttered side down, in a large skillet set over moderate heat. Cook until the bottoms are crisp and deep brown, about 3 minutes. Remove from the pan and repeat with the remaining bread.
Put 4 of the cheese toasts on a work surface, browned side down. Arrange an avocado half and 3 tomato slices on each toast and season with salt and pepper. Top each with 1 onion slice, 1 tablespoon of the mustard-mayonnaise and 1/2 cup of alfalfa sprouts. Top with the remaining cheese toasts, browned side up.
The vegetables would suggest a light white as a flavor foil, but the cheese gives the sandwich enough richness to pair with a round, fruity California Chardonnay, such as Meridian Santa Barbara or Alexander Valley Vineyards Wetzel Family Estate.