How can I melt mozzarella balls?

Very simply, with heat from above.

I cannot comment on the microwave, since I have almost never used one, but a pan is clearly the wrong tool for the job, since there you are heating the bread from below, and the cheese only gets slightly warmed, if at all.

Using an oven, I have never noticed a need for reducing the moisture, as other answers suggest. The mozzarella straight out of the brine might make your bread somewhat soggy, but there is nothing wrong with that for me.

The best device for doing this will be a grill/broiler, but a toaster oven and a normal oven will also work. All you have to do is slice the mozzarella, place it on the bread, and bake. Place the sandwiches as closely to the upper heating element as possible.

Using this method, I have made sandwiches with up to 1.5 cm thick mozzarella slices. They are quite decadent :) but my point is, the thickness is not a problem at all, nor is the moisture.

Above, a picture of some mozzarella I melted on bread in an oven. It is the high-moisture mozzarella sold in apple-sized balls, sliced thickly. I placed it as close as possible to the upper heating elements, and it fared quite well.

Mozzarella balls, particuarly the almost "Fluorescent" white variety, contain a lot of water. To get them to melt successfully, you need to get that water out.

What I do is slice the balls into ~ 5mm rounds and leave them to drain in a fine sieve for about an hour. You could also leave them in the fridge overnight, this would dry them out even more. You could try not slicing them, but I think they would still retain a lot of water that way.

Using this method I can get melted mozz on my pizza from pale white to nicely browned in a regular domestic oven at 250C [482F].

First off your best bet is going to be an oven/air fryer but a pan should also work. Secondly, you are probably using the wrong type of mozzarella. What you want for things like pizza and other melting tasks is low-moisture, whole milk mozzarella. Low moisture melts much better, and you also want high fat content whole milk provides. It might be harder to find this, but it's definitely worth it. Some packaged 'string' cheeses can be low moisture with higher fat, so don't forget to check them too.

"Real" Pizza tends to have the balls broken up or sliced and done under intense heat.

I've had decent luck getting fresh (balls in brine) mozerella to melt in a toaster oven simply by breaking it up into fairly small pieces so they have more surface area to heat up.

If you're doing it on a pan, you could try melting the cheese on a pan that's hot but not on direct heat and build your toast 'upside down'

A method for "pan on stovetop" but not really for pizza (very effective at cook top and melt cheese, but not in a crispy pizza like way) is to put a lid on the pan (which essentially steams the top of the food, thus the disclaimer that it's not pizza-like, but does melt cheese effectively.)

Handy to know if you are stuck with no oven.