In a small microwave-safe jug, pour in the water and milk, then add in the sugar and stir to mix. Place the jug into the microwave and zap it for 30 seconds. Stir the mixture and place a thermometer in. The mixture should be between 36.5 to 40.5 C - 100 to 110 F. You may need to leave it to cool for a few minutes if it's too hot. Once it reaches the correct temperature, stir in the yeast. Cover and leave for 7 minutes or until foamy.
Place the flour and salt into a stand mixer bowl. Use a dough hook and operate at low speed (KitchenAid on level 2). With the motor running, add the egg into the flour. Give the yeast mixture a mix and drizzle it into the bowl, then drizzle in the melted butter. Once the dough comes together and moves away from the sides of the bowl, leave the mixer to knead the dough for 5 minutes.
Lightly grease a large bowl with oil (I used oil spray). Gather the dough and make a ball. Place the smooth side of the dough down into the bowl to grease, then turn it over. Cover the bowl with a tea towel or plastic film. Leave the bowl in a warm and draft-free place for 30 minutes or until it's doubled in size.
Lightly flour the workbench with flour and hand knead the dough for 3 to 5 minutes. The dough should feel soft but slightly elastic. Shape the dough into a rough rectangle and roll it into 40x22cm (16x9in). Don't worry if you can't get a perfect rectangle shape because we'll be trimming off the ends later.
In a small microwave-safe bowl, add in the vegemite and butter. Zap it in the microwave for 10 seconds until the butter has melted, then mix well to combine.
Brush or spread the Vegemite mixture with a flat spatula all over the dough, then sprinkle the shredded cheese on top. Tightly roll the dough lengthwise until it reaches the other end, turn the dough so the seam faces down on the bench.
Lightly grease the bottom and sides of a 23x33cm (9x13in) with oil spray. Slice off the uneven edges of the roll and cut the roll in half, then cut out 12 scrolls in total. The rolls may have flattened slightly after slicing, so use your hands to help make them round again. Place the slices into the tin and spread them out evenly. Cover the tin and let them rest again for 1 hour or until they've doubled in size.
Preheat oven to 210°C (410°F) or 190°C (370°F) for fan-forced ovens when there are 15 minutes left remaining for the dough to rest. Bake the scrolls for 15 to 18 minutes or until golden brown on top. Leave them to rest in the tin for 10 minutes before serving.