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Tip the beef mince, shallot, horseradish sauce, herbs and garlic into a bowl, and season well with salt and black pepper. Combine everything well using your hands, squeezing the mixture through your fingers repeatedly to help it bind together.
Divide the mixture into four portions and shape each into a 3cm-thick patty. Press a small 50p-sized piece of blue cheese into the middle of each patty, if using, and pack the meat around it to enclose the cheese – this will melt during cooking.
Put the burgers on a plate, cover and chill for at least 30 mins or overnight. Light the barbecue and wait for the coals to turn ashen. Or, heat a gas barbecue to medium.
Lightly oil the burger patties all over and cook for 5 mins until just starting to char. Carefully flip over and cook for a further 4-5 mins (for medium) or longer if you prefer well done. Remove the burgers to a board and leave to rest for a few minutes before serving. Alternatively, fry the burgers over a medium heat for 4-5 mins on each side until cooked through (a probe thermometer should read 70C).
Serve the burgers in the buns with lettuce, tomato slices and relish (see recipe, below), or any other accompaniments of your choice.