Using a balloon whisk, mix the flour and water together in a pan until smooth.
Place the pan over a low-medium heat and stir constantly with a spatula until thickened to a pudding-like consistency. If you have a thermometer the paste should reach 65°C [149°F] before you take it off the heat.
Transfer the tangzhong to a small bowl, cover with plastic wrap (making sure this touches the surface of the tangzhon) and chill in the freezer for 10 minutes.
Warm the milk in the microwave. It should be warm but not hot. Melt the butter, too.
Make the dough
Lightly oil a large bowl and grease a 20-cm (8-in) square baking tin, then set aside.
Place the milk and butter in a large bowl and add the sugar and salt.
Add the chilled tangzhong to the bowl along with the egg and whisk together.
Add the flour and yeast to the mixture.
If using a stand mixer, just allow the machine to knead for 10 minutes with the dough hook attachment. If working by hand, use a wooden spoon to combine everything into a shaggy ball of dough, then turn out onto a floured surface.
Knead by hand for about 10–15 minutes. The dough will be sticky to start with, but avoid adding too much flour – it will gradually become less sticky as you knead it. If the dough sticks to the surface, use a dough scraper to scrape it off. Keep kneading until the dough is smooth; it will still be a little tacky, but that is normal.
Place the dough in the oiled bowl and cover with plastic wrap. Leave to rise in a warm place until doubled in size. This takes about 1 hour, but it depends on the temperature of your kitchen.
When the dough has doubled in size, turn it out onto a lightly floured surface and knock back.
Form into 9 balls, weighing 65g (2¼oz) each. Shape them by tucking the dough under to create a smooth surface. There will be leftover dough – this is to shape the ears and paws.
Shape the small balls of dough into ears and paws for the cats. You can brush on a little milk to help them stick. Shape a tail for one of the buns – this will look like the cat has turned around. Work fast to avoid the dough forming a skin.
Place the dough balls in the prepared square tin. Cover with lightly oiled plastic wrap and leave to prove in a warm place until roughly doubled in size. The time this takes varies depending on the temperature of your kitchen, but won’t be as long as the first rise. You are looking for the dough to have nearly doubled in size and spring back halfway when lightly pressed with a finger.
About 15 minutes before the end of the proving time, preheat the oven to 180°C (350°F/Gas mark 4).
Brush the beaten egg on top of the buns just before baking.
Bake for 20 minutes until lightly golden brown. You will need to cover the buns with foil after 5–10 minutes, just to prevent them browning too much.
Transfer to a rack and leave to cool completely.
Paint coloured patches on the cold buns using a little brown food dye mixed with a tiny bit of vodka.
When dry, use black edible pen or royal icing dyed black to add cute facial features and details to the paws.
Once you've mastered this recipe, you can try decorating the dough as other animals. How about mouse, dog or even guinea pig themed buns!