In a bowl, whisk 32 ounces plain whole-milk yogurt with 1 teaspoon fine sea salt until smooth.
Line a fine-mesh sieve with a double layer of cheesecloth and set it over a deep bowl. Scrape the salted yogurt into the lined sieve and fold the cloth over the surface.
Refrigerate and drain until thick and spreadable, 12–24 hours, emptying the collected whey once or twice as needed. The labneh is ready when it holds soft peaks and no whey drips when spooned.
Transfer the labneh to a clean container and stir to smooth. For a firmer cheese suitable for shaping into balls, continue draining up to 48 hours.
To serve, spread the labneh in a shallow bowl and drizzle extra-virgin olive oil (for serving) over the top. Refrigerate leftovers in an airtight container for up to 1 week.