Just some math behind the frames and timing, for consideration.
Assuming you are running standard deep frames for the brood. 3500 cells per side, 7000 cells per frame. The bees are going to fill and use 30%-40% of that for resources. Nectar pollen, bee bread, cleaning/polishing. Leaving 0.6*3500; 2100 per side, 4200 cells per frame available to the queen to lay in. ( Hey! Jim look .... -42- )
A decent average queen will lay 1200-1400 a day, in good balmy weather and a busy buzzy rockin it out hive. Some will do more some will do less. If she is a good old queen, then let us just use 1200 as her lay rate. Based on that it will take the queen 2100/1200 days per side or 4200/1200 days per frame to layout. ( 1.75 days per side, 3.5 days per frame )
The bees, assuming there are more than one of them on the frame working it, will certainly be able to prepare and polish the cells faster than the queen will lay them. So you can totally disregard the thought or concern about there being nice polished unused frames in the other hive body. It is not worth the effort or the disruptions to the nests to go shuffle those. Leave the bees to do what bees do with the combs.
If you would like to use her in the split as a brood factory, and for grafting new queens, that is a wonderful idea. Ensure they have a constant supply of pollen and feed without having to do much foraging. Watch their stores level and if necessary feed the split she is in.
As for when to harvest frames, the math is above. Once per week you can pull 2 frames of brood out of her. She may and will lay out just a bit more than 2 frames per week, but leave that extra bit with her. Old bees do not raise brood well, and old bees die. So she also needs her bees in the split constantly being replenished with new bees as the old ones die off. A mistake folks make with brood factories is to pull all the brood each time, each week, then wonder why the brood factory crashed. Simple, she needs brood too! So ensure to leave her some.
Once a week pull 2 brood frames. Before you go stick the frame in a separate hive, graft a few larvae off the frame into cups to start some queen cells. Put 4 to 6 of your grafted cups between the centre top bars of each split. Then go put the frame into a hive that needs the boost. Adding the grafting step gives you some practice and gets the splits drawing some cells into a place that you can move them if need be. You can move the extra drawn QC cups from the successes to other splits that failed to start any. It adds some fun factor and allows you to manage the cells easily without having to pull frames and cut cells out of the combs. Just a thought.
Three important points / prerequisites. 1. She has enough bees to keep up with her to work the frames and raise the brood, and 2. there is copious amounts of feed (protein) right next to them. 3. Do not take all the brood each week. Ensure she is left a half frame of emerging brood each week.
Hope that helps!