Lightly oil your mixing bowl for proofing dough later.
In the stand mixer bowl, stir the warm milk, yeast, and about a teaspoon of the sugar together. Let this sit for 5-10 minutes, until it's foamy.
Add the rest of the sugar, the egg and egg yolk, salt, and melted butter to the yeast and start mixing with the dough hook. Start at medium speed to incorporate it all.
Turn the speed to low and start incorporating the bread flour a few spoonfuls at a time. Keep this going until all of the flour has been added, and let the mixer knead it for an additional 10 minutes. Optionally, hand-knead for a few minutes to get a feel for it.
Transfer the dough ball to your oiled bowl and cover it with a clean towel. Let it proof at room temperature for around 1½ hours, or until it has doubled in size.
Grease your baking pan with some extra butter and line it with parchment paper for easy removal of the rolls when they're done.
Once your dough has risen, transfer it to a well-floured surface. Roll it out using the rolling pin. You want to get it to be rectangular, about 9x14".
Using a spatula or knife, spread the softened butter across the surface. Leave one of the short ends unbuttered to better seal the rolls later.
Mix together the brown sugar and cinnamon until it's crumbly and sandy. Spread this mixture across the buttered surface of the dough. Spread the strawberries across this evenly.
Starting with the short side, tightly roll the dough into a spiral, careful not to squeeze out the filling as you go. The unbuttered edge should be the last bit rolled up.
Optional: If you're using a small pan, or your dough wasn't a nice rectangle, cut about half an inch off of each side, since these edges will be uneven and poorly filled. We didn't do this, and two of the rolls ended up pretty misshapen and empty.
Measure your dough roll, and mark out 8-10 even pieces. If you're using a square 9x9" baking dish, cut nine; if you're using a larger rectangular dish, cut either 8 or 10 depending on how many and how large you want.
Arrange your freshly-cut rolls in your pan. Cover this with the towel and let it rise again, for at least 30 minutes. You can let it rise overnight in the fridge, but make sure to let it come to room temperature for baking.
Preheat your oven to 350℉. If you want, brush your rolls with the egg white for an egg wash, but you don't need to. Bake the rolls for 30-40 minutes, or until nicely golden. A thermometer showing the bread at 185℉ is a good way to tell they're done.
Remove form the oven and let them start to cool while you make the frosting. To make the frosting, add all of the ingredients to your stand mixer with the paddle and whip it together until it's fluffy. Spread over the cinnamon rolls and serve!