Right, so first off to just get on the right page...they lay eggs...the eggs hatch...Dragonborn chicks got melons.
Okay, hows this..
Yes, post hatching, they do breast-feed their young. The kids DO mature much faster, but they also likely require a LOT more calories and nutrients due to the rapid development. Whats a year or two in normal children they accomplish in 3 months or so. All the fuel for that development has to come from somewhere, and while starting junior out with half a cow to chew on when he gets hungry may work for true dragons, Dragonborn are somewhat humanoid as well, and humans require alot of nutrients and the like for the best development. Things that maybe other sources cant give them. THUS, dragonborn women have breasts for breast feeding their youngsters because the milk is likely a far better source of nutrition for them than an entire bucket of cow or goats milk every few hours. Its also likely the consistency of mud...but thats just me thinking this out just a bit TOO far...
The trick isnt to look at them as lizard people, but rather to look at them as People with reptilian qualities. Conception is likely mammalian cause everyone knows hot kinky lizard person sex is all the rage these days. The birthing is reptilian, cause the dagonborn women have to watch their girlish figure. Nutrition is mammalian cause hot kinky lizard person sex just isnt the same without boobs, and growth and instincts are somewhat reptilian, because they develop so quickly.
So yeah, im a sad sad person for thinking that all the way out. Im even more ashamed that I went and joined the forum purely so I could throw this out there to be read. Ill just go back to lurking now.