In the bowl of a stand mixer fitted with the kneading hook, whisk together the milk, sugar, and salt. Add the egg and whisk to incorporate. Whisk in the instant yeast. Using a fork, gradually incorporate the flour into the wet ingredients, then start kneading the dough. Gradually incorporate the butter into the dough as it is kneading. Keep kneading the dough, adding a bit more flour if it is too sticky. After about 7 minutes, the dough should be smooth and elastic.
Transfer the dough to a large and oiled mixing bowl and cover with a clean kitchen towel. Let rest in a warm spot for about 1 hour, or until the dough has doubled in size.
Grease two loaf pans, then line with parchment paper. Set aside.
Roll the dough out into a large rectangle, about the size of a baking sheet. Brush the toasted sesame oil all over the dough, then sprinkle with the salt. Distribute the shredded Armstrong Mild Cheddar cheese all over the dough, then add the minced green onion.
Roll the dough up from the long side of the rectangle until you get a log. Slice the log in half to create two shorter logs. Set one log aside.
Using a very sharp knife, cut one log right down the center, lengthwise. Lay both sliced pieces side-by-side, open sides up, then press the tops together and twist them around one another. Transfer to one of the prepared loaf pans. Repeat to prepare the second babka.
Transfer the loaf pans to a warm spot, cover with a clean kitchen towel, and let rest for 40 minutes more.
Preheat the oven to 175°C (350°F). Bake the babka for about 30 minutes, or until the bread is puffed and golden brown and sounds hollow when you tap on it.
Let cool slightly on a wire rack, then unmold, slice and enjoy warm.