Combine mango slices and 3 tablespoons salt in a large bowl. Let sit, about 15 minutes.
Combine remaining 1 tablespoon salt, 3 cups plus 3 tablespoons water, and sugar in a large pot; bring to a boil. Stir until sugar is dissolved. Remove from heat; pour in vinegar. Remove brine from heat and let cool, about 15 minutes.
Drain mango slices and place in large jars. Cover with cooled brine. Seal and refrigerate until mango turns yellow, about 2 days.