Measure ½ cup milk into a 2-cup microwave-safe measuring cup or Pyrex bowl. Whisk in 2 tablespoons unbleached all-purpose flour Cook on High in a microwave for 45-60 seconds (microwaves vary) whisking every 15 seconds until thickened like pudding.
Dough
Add ½ cup cool milk to the Tangzhong paste (cooked milk/flour mixture from above) and whisk energetically. It's OK if there are still some lumps. They will disappear in the mixing process. Add to the bread machine pan.
Add1 large egg ,1 tablespoon heavy cream OR 1 egg yolk, 2 tablespoons sugar , 1¼ teaspoon salt , 4 tablespoons butter, 2¾ cups unbleached all-purpose flour , and 2 teaspoons instant yeast to the bread machine pan and select the DOUGH cycle. Press "Start."
During the first minute, lift the lid to make sure the paddles are engaged and working correctly. Dough should start to clump and begin to form a ball.
Recheck 12-15 minutes later to ensure the dough is the correct consistency. It should stick to the side, then pull away cleanly. If the dough is too dry it may thump against the side of the pan or ride around freely on top of the post. Add milk one tablespoon at a time. If the dough is too wet, impossibly sticky and won't pull away from the sides, add flour one tablespoon at a time. Give the dough time to absorb the flour or milk between each addition.
When the DOUGH cycle finishes, check to see if the dough has risen to double its original size. If so, remove it to a floured surface. If not, allow the dough to remain in the machine until it is doubled.
While the machine is doing all the hard work for you, grate the cheese so it has time to come to room temperature.
Making and Baking the Rolls
Remove the dough from the bread machine onto a floured surface. I like to use a silicone mat because clean-up is easy (dishwasher).
Divide dough in half. Form each half into a neat ball, flouring the surface as you work to prevent sticking.
Use a rolling pin to roll one dough ball into a 12 x 9-inch rectangle.
Evenly distribute half of 2 cups Cheddar cheese over the top of the rectangle.
Divide into 4 equally-sized strips lengthwise, then divide each strip in half across the width. You should have 8 total strips. Roll each strip starting from the outside ends.
Place rolls into a greased (spray with Baker's Joy for best results unless using USA pans) 8 or 9-inch pan in a spoke fashion.
Repeat the shaping instructions with the other half of the dough.
Cover the pans with tea towels or cheap shower caps and set aside to rise again until almost double. This will probably take an hour, plus or minus, depending on the room temperature.
Preheat oven to 375 °F (190˚C) when you think the rolls are almost ready to bake.
Place rolls onto the middle shelf of your oven. Bake for 14-17 minutes or until golden brown and the internal temperature reaches 190˚F (88˚C).
Remove from the oven and allow to sit for about 5 minutes. Run a plastic knife around the edge to loosen any cheese stuck to the edge. Turn rolls out onto a wire rack for another 10-15 minutes. Their tops should be up so their shape is not affected. Brush all over with melted butter.
Tip: If you leave rolls in the pan too long after baking, they will sweat and become soggy on the bottom.
Notes
Directions for making bread with a stand mixer or by hand:
To make this recipe in a heavy-duty stand mixer: Add ingredients to the bowl in the same order. Turn on LOW to mix until all ingredients are moistened. Then, using a dough hook, turn the speed to 2 or 3. Continue beating/kneading until dough becomes smooth and elastic (about 5-10 minutes). Cover and allow to rise in a warm place. Deflate dough gently and shape as indicated in the recipe.
If making by hand: Combine all ingredients into a shaggy ball in a large bowl. Turn dough out onto a floured surface. Knead with your hands until the dough becomes smooth and elastic. Kneading will likely take 10-20 minutes, depending on your experience. Place the dough ball into a greased bowl. Cover and allow to rise until double. Deflate the dough gently and shape as indicated in the recipe.
Please note: If you only have active dry yeast, use 1/4 teaspoon more than called for in the recipe. It no longer needs to be dissolved first, but you can if you prefer.