Heat 1 tbsp butter and half the oil in a large heavy-based frying pan over a medium heat, and fry the onion with a pinch of salt for 20-25 mins until golden and sticky.
Tip the mince into a bowl with the garlic granules, Worcestershire sauce and some seasoning, and combine with your hands – mix until the mince has a fine texture. Divide into two balls, then flatten these out into patties, keeping the shape of the bread in mind (you may want to make them more oblong if using a bloomer loaf).
Scrape the caramelised onions onto a plate, and heat the remaining oil in the pan over a high heat. Fry the burger patties, pressing them down with the back of a spatula, for 3-4 mins on each side until golden brown and cooked through. Transfer to a plate, scraping any bits from the pan over the top.
Reduce the heat to medium-low, and melt ½ tbsp butter in the pan, swirling it around the base. Fry all four of the bread slices on one side, weighing them down with a grill press or pressing down with the back of the spatula so they’re evenly golden and crisp. Do this in two batches, adding another ½ tbsp butter for the last two slices. Mix the two cheeses together in a bowl.
Put two of the bread slices, toasted-side up, on a board, and spread over the mustard. Add half the cheese, and spoon over the onions evenly. Top with the burgers, then scatter over the remaining cheese. Top with the remaining bread slices, toasted-side down, so the untoasted sides are on the outside of the sandwiches. Press down well to seal. Heat most of the remaining butter in the pan, and fry the sandwiches for 2-3 mins, pressing them down the whole time, until golden and toasted. Carefully flip and fry again, adding the last of the butter to the pan around the toasties. Cook until the cheese is melted and oozy, and the bread is golden and crisp on both sides.