Put the lamb in a bowl. Grate the onion (medium-fine grate), finely grate or crush garlic. Add both, along with the parsley, mint, oregano, cumin, pepper and salt to the lamb. Mix well until all the seasonings are well distributed (often best by hand). Divide the mixture into 4, roll into flattened balls and indent in the middle. Place on a baking sheet/tray and chill until needed.
Halve or quarter the tomatoes (or dice if using larger tomato) and finely chop the onion. Finely slice the mint. Mix them together with the oil and vinegar and set aside. Pre-heat the grill.
Crumble the feta and mash or blend it up with the yogurt, oil, lemon and honey until relatively smooth. Chill until needed.
Cook the burgers on a pre-heated grill over a medium flame, or cook on a grill pan/in a skillet. Watch if they catch in the grill flame - move away from direct heat if you need to. Cook for a few minutes each side until cooked as you prefer - medium is my preference, so there is a slight 'give' when pressed but not too soft.
Lightly toast the buns, put some feta sauce on one side of each and top with the burger, some arugula and some of the minted tomatoes.