In a large pot, heat a gallon of water to a simmer. In a large bowl, break up the mozzarella curd into small pieces. Coat the curd with 1 tablespoon salt, and set aside. In another large bowl, add remaining salt to the second gallon of water along with as much ice as it can hold.
This is where it gets a little tricky. Carefully pour the hot water over the salted curd. As the curds begin to melt, carefully stir with a wooden spoon or spatula for 1 minute. Submerge both of your hands in the ice water for as long as you can stand it, because you're about to stretch the mozzarella in hot water. To protect your hands, you may wear rubber gloves if desired.
The curd will have settled into the bottom of the bowl, and at this point you have to work quickly or risk over-cooking the curd and it becoming rubbery. Dip your hands into the hot water and begin stretching the mozzarella. By stretching the cheese, you insure that all the curds have completely melted. Form the mozzarella into tomato size balls, and then drop the mozzarella into the ice water to set.
Slice the mozzarella in 1/4-inch slices. On a serving platter, arrange mozzarella and tomato slices, alternating and overlapping a bit. Top with basil. Drizzle with olive oil; season salt and pepper.
Wrap remaining mozzarella balls tightly in plastic wrap and store in the refrigerator for up to a week.