Put half the flour (just eyeball it), the oats and the beer in the bowl of your stand mixer. Mix until well-combined, then cover and let rest for 30-45 minutes to soften up the oats.
Mix the milk and yeast together and add that to the bowl along with the rest of the flour, the salt, molasses, melted butter and sugar.
Mix on low speed using the dough hook. The mixture will look pretty wet, but forge ahead.
Once all the ingredients are mixed, turn the speed up to medium and knead until you have a sticky dough that clears the sides of the bowl but sticks in the bottom of the mixer bowl in about a 2" circle. Knead for about 10 minutes. The dough will be very soft, and you'll have to oil hands or coat them with pan spray to work with it.
Form the dough into a reasonable-looking ball and drop it back into the mixer bowl. Spray the top with pan spray. Cover and let rise in a warm, moist place until doubled, about 2 hours. I bring a mug of water to a boil in the microwave and then scoot it to one side and let the dough rise that way.
Once the dough has risen, gently press out the gases. Divide the dough so that one piece that weighs 1 1/2 pounds. With oiled hands on a lightly oiled surface, press that piece out into a rectangle and then roll it up into a log. Fit it into an 8 1/2" x 4 1/2" baking pan, spray with pan spray (or brush with a bit of oil) and cover.
With the remainder of the dough, roll it into a smooth ball, tightening the top as much as possible. Place on a sheet of parchment, spray with pan spray or brush with some oil and cover. Let both loaves rise in a warm, moist place until almost doubled again, about 90 minutes.
Preheat the oven to 400F a good 45 minutes before baking. Place a baking stone on a rack one down from the center of the oven.
When ready to bake, gently but thoroughly brush the tops of the loaves with the egg-milk mixture and then evenly toss on some rolled oats.
Slide the round on the parchment onto the baking stones and place the baking pan next to, it, leaving enough room so the round doesn't bake onto the pan.
Bake for 10 minutes.
Reduce the heat to 350F and continue baking until the loaves are a lovely deep golden brown and the internal temperature is between 195F and 200F. You may have to cover the loaves with foil to prevent overbrowning, so keep an eye on them.
My round was finished in about 30 minutes while the loaf in the pan took around 40 minutes.
Let cool on racks, turning the loaf in the pan out after about 5 minutes so it doesn't get soggy. Let cool to below 140F before slicing or ripping off a chunk and devouring. If you can wait that long.