Im not sure exactly how long it would take, but temperaments are bread in to all other kids of animals by selective breeding, I don't know why you couldn't do it with beardies. My thought is, the stronger the gene, the easier it would be to get an easy going beardie in a brood. And if you have two beardies with a dominant "gentle" gene, then it seems it should work.
Selective breeding has been done in animals and plants by humans for a long time now. Darwin studied selective breeding (also called artificial selection). Selective breeding is also done in nature. But in other domesticated animals such as dogs and cats; temperament, color and size have all been enhanced by selective "artificial" breeding to get the desired outcome in breeds. I don't know why it can't be done with BDs as well… Of course, as with all other animals, there will be recessive genes that come out in any selective breeding experiment, and in a clutch by the two beardies any variety could happen. But the chances are greater and greater for more docile babies with every generation. So, it seems it would depend on how far down the gene pool both of your current beardies have had a dominate nature to be docile, and gentle, and accepting of dependence on humans.
When we purchased our BD, he was housed with his clutch brother, and although they were in the same tank for the first 4-5 months of their lives, they seemed completely different from each other. His brother was skittish around people, while Rin was curious and wanted to be held. We chose him based on personality alone. He is now around 8-9 months old and is extremely sweet and gentle natured. He has not yet
black bearded anyone. We took him back to the reptile shop where we purchased him, over the weekend, and showed him several other beardies of all ages (from babies to 5 year olds), and he just sat in my hand with all of them, curious, and following them with his eyes but never seemed threatened nor threatening to them. He also saw a lizard that puffs up the side of his face when he gets threatened. As soon as we lifted Rin to that viv, the other lizard swelled his face up and went in to either attack or vulnerable mode by flaring out his flaps on his face, but Rin just sat there as if he was watching a rock. I mention all of this because he was MUCH different than his clutch brother a few months back when we got him. The difference was night and day between two clutch brothers. So, I'm sure there will be the same variables in your clutches, even with two very docile and gentle BDs as their generational parents. But I'm thinking the odds of MORE of the clutch being like both parents in demeanor is greater than not.
I mentioned on an earlier post that my 9 year old daughter came up with the concept that, even though BDs seem to be solitary in nature, as they are bread more in captivity, it's easier to breed two BDs that are easy to handle and more easy to be dependent on humans, so the more captive breeding, the more dependent (hence seemingly easy to bond with humans) they become.
I find the entire concept fascinating, and that any animal which is independent if left to nature, and not a pack animal, of sorts, could be willingly dependent and part of the "pack" in a family so easily.