Using a box grater or food processor, grate the whole cucumber and then place it in cheesecloth or a tea towel. Squeeze out as much liquid as you can from the shredded cucumber, ensuring your sauce remains thick and creamy.
In a large bowl, combine the drained cucumber with plain full-fat Greek yogurt, minced garlic, extra virgin olive oil, lemon juice, and salt. Stir well until all ingredients are evenly mixed together.
Cover the bowl with plastic wrap or a lid and refrigerate for at least 2 hours, allowing the flavors to meld together perfectly. For optimal taste, refrigerate overnight if possible.
Once chilled, give your tzatziki sauce a quick stir before serving it alongside pita bread, fresh vegetables, grilled meats, or gyros. Enjoy immediately or store in an airtight container in the refrigerator for up to five days.