Peel cucumber and roughly chop. Sprinkle them with salt, set in colander and place a heavy item on top to draw out water. Let sit for 30 minutes, then drain well and wipe dry with paper towels.
Add cucumbers, garlic, lemon juice, dill and a sprinkle of pepper in a blender or food processor. Blend the mixture until smooth, then stir in Greek yogurt. Add in salt to taste. Place tzatziki in refrigerator for at least one hour to ensure the flavors are blended.
Set sous vide machine to 57C/135F.
In a bowl, mix ground lamb, dried oregano and salt and pepper until well blended. Divide mixture into four equal portion and shape each into a patty about 4 inches wide.
Put the patties in a freezer bag, and remove the air through a vacuum sealer or the displacement method. Drop bag in the bath for 45 minutes.
Remove patties from bath and pat dry with paper towel. Turn grill on high and cook the patties for 1-3 minutes on each side. Alternative: If you do not have a grill, heat pan on high, add oil or butter and sear the patties for 1 minute on each side.
Place patty on warm, toasted bun and top with homemade tzatziki sauce, fresh arugula, red onion and your other favorite toppings.