When I do a split that is not from a hive that swarmed, I make sure the one that doesn't have the queen gets a frame with eggs on it. Never had a problem.
The split from a swarmed colony get capped queen cell.
The both colonies that I split that had swarmed had several capped cells. I divided these up between the splits. I have laying queens. 3 with new laying queens and the mother colony of one still has the old queen.
The hive that swarmed on April 16, was split on the 22 and swarmed again 23.
On 5-14 I checked for a queen and couldn't find, queen, eggs or brood.
On 5-22 there was eggs and young brood, I did not look for the queen.
I know how it is to want to see the queen, but if you find eggs and brood and you know it is not working layers, then you are just disturbing the bees more than need be and taking a chance on injuring or killing the queen.