To make your own buns, tip the flour, sugar, yeast and 1 tsp salt into the bowl of a stand mixer fitted with a paddle attachment. With the motor running on medium speed, gradually add the milk, then half the egg. Leave to mix for a few minutes, then add the butter, a little at a time, until the dough is elastic and shiny – about 5 mins. Alternatively, do this in a bowl using an electric whisk.
Divide the dough into four and roll each piece into a round on a lightly floured surface. Arrange well spaced apart on a baking tray lined with baking parchment. Press down gently to spread them out a little and leave in a warm place for 2 hrs, or in the fridge overnight, until doubled in size.
Heat the oven to 200C/180C fan/gas 6. Brush the remaining egg over the buns and scatter with the sesame seeds, if using. Bake for 13-15 mins until risen and golden. Leave to cool. Can be made up to a day ahead.
For the sauce, stir together all the ingredients and set aside. Can be made up to a day ahead and kept chilled.
To make the burgers, chill the chuck steak in the freezer alongside the blade and, if it fits, bowl of a food processor for 30 mins until the meat is firm but not frozen solid. Working quickly, tip half the chilled beef into the food processor and pulse eight to 10 times until the meat is minced, being careful not to overwork it. Tip the mince onto a tray and repeat with the remaining steak, then mix both lots together with ½ tsp salt. Divide into four portions and gently form each into a loose patty a bit wider than the buns. Don’t overwork it.
Cook the burgers as soon as possible after forming. Set a large, heavy frying pan over a medium heat. Split the buns and toast, cut-side down, in the pan (you may need to do this in batches). Remove to a plate, then sear the burgers, pressing them down with a spatula as they cook, for 4-5 mins on each side for medium rare or 6-7 mins for medium. Top with cheese, if you like, after flipping them. When the burgers are cooked, assemble as preferred with the buns, toppings of your choice and the burger sauce.