Combine all the ingredients for the green sauce. Season to taste and leave to one side until needed.
Rub the meat all over with olive oil and season with thyme, seas salt and black pepper. On a griddle or a barbecue, sear the loin for approximately 4 -6 minutes on each side, or until you think it is right. It should be well coloured on the outside and pink within. Allow to rest for 8 minutes or so. This is important as, during this time, the meat will finish its cooking and become relaxed and tender.
Toast the bread ( preferably on a barbecue) and spread half the slices with the mustard.
Slice the venison and lay it on top.
Apply lots of green sauce and the remaining toasted slices and eat the sandwiches immediately.