Toss tomatillos, kosher salt, and sugar in a small bowl. Gently massage until they feel soft and juicy. Cover and let sit at room temperature, tossing occasionally, until tomatillos have released some of their liquid, at least 4 hours and up to 12 hours.
Preheat oven to 300°. Holding a bread knife parallel to a cutting board, slice ciabatta in half. Slice each piece in half lengthwise, then cut each strip on a diagonal into about 4" pieces. Drizzle 3 Tbsp. oil over bread and rub each piece to evenly distribute oil. Place bread on a baking sheet and bake until browned all over and dried out and crisp, 60–75 minutes. Let cool.
Meanwhile, transfer tomatillos to a food processor; leave juices in bowl. Pulse until finely chopped but not smooth (mixture should look pulpy with bits of skin). Transfer to a medium bowl; add garlic.
Spoon tomatillo mixture over toast, then drizzle some of the reserved juices over. Top with green tomatoes, if using; drizzle with oil and sprinkle with sea salt.