Trim the ends of the plantains and cut them in half crosswise. With a sharp paring knife, score the skin along one or more of its ridges, being careful not to cut into the flesh, and then peel off the skin in sections. Cut each half into 1-inch rounds.
Heat a 10-inch skillet with 1/2 cup of the oil over medium heat. When the oil is hot (a plantain will sizzle), put five plantain pieces in the skillet; don’t crowd the pan. Fry until light golden on both sides, 3 to 4 minutes per side. Transfer the pieces with tongs or a fork to a plate lined with paper towels. Cook the remaining pieces the same way. This initial fry softens the plantains so they can be smashed.
Remove the pan from the heat but don’t pour out the oil. Put a plantain round between pieces of plastic wrap. With a smooth meat pounder or the bottom of a glass, gently smash the round so it flattens but doesn’t crumble; it will have uneven, ridged edges and be about 1/4-inch thick. Smash the other plantains in the same way.
In a large baking dish, dissolve the salt in the warm water. Fully submerge as many smashed plantains in the dish as will fit and soak for 3 to 5 minutes, but no longer. Dry the plantains well between paper towels.
Add the remaining 1/2 cup oil to the reserved oil in the pan and heat on medium high (a plantain will bubble around the edges when added). Fry four or five tostones on each side until golden, 2 to 3 minutes per side. Transfer to a platter lined with paper towels and immediately sprinkle with kosher salt. Continue frying the remaining rounds. Serve hot.
To make a dipping sauce for the tostones, sauté 3 chopped garlic cloves in 3 Tbs. olive oil, and then combine with 1/3 cup fresh lime juice, a dash of ground cumin, and salt and pepper to taste.