There's no point making your own burgers if you don't have decent buns to put them in. The perfect burger bun needs to be soft enough to bite into without breaking your teeth, but strong enough to hold together while you consume the feast inside.
Combine the flours in a bowl, add the salt, crumble in the yeast and stir them all together. Add the egg, butter, milk and water and stir well. Tip the dough out onto the oiled work surface, and knead for 10 minutes until the dough is smooth and elastic. You may need to add a bit more flour if the dough is too sloppy. To knead it, roll it into a sausage, fold it over in thirds, turn through 90 degrees, stretch into a sausage again and repeat the folding, turning and stretching.
Roll the dough into a ball and put it in the bowl. Cover with plastic film and leave to rise in a warm place until doubled in size - this should take about one hour.
When the dough has risen, tip it out onto the work surface and flatten it to knock the air out of it. Knead it again for 10 minutes and then divide into 4 equal pieces. Shape them into balls and place them on an oiled baking sheet to rise. I usually put mine into oiled steel rings to get a nice circular shape and a good vertical rise. Cover them with a clean cloth and leave them to rise again until doubled in size.
When they have risen, preheat the oven to 200℃ (392℉). Brush the top of each bun with beaten egg and sprinkle on some sesame seeds. Place a tray of boiling water in the bottom of the oven, and put the buns in to bake on the middle shelf for 15 minutes.
To check for doneness, tap the bottom of the buns - if it sounds hollow, it's done. Remove from oven and place on a wire rack to cool.