The fact that Beast Boy is able to lay an egg with a chick in it may be an example of parthenogenesis, the process by which an unfertilized egg develops an embryo.
Also, the fact that Beast Boy appears to be a rooster when he lays the egg may not be so strange. At times, chickens take on rooster-like qualities, usually because of a hormone imbalance.
If you observe the cupboard in the living room carefully during the scene of Beast Boy taking out a baseball glove, you’ll notice that Pain Bot was inside with its bulb broken.
It’s possible that it got damaged after Beast Boy threw it out of the tower in “TV Knight”.
This isn't Beast Boy's first chicken son. In "Pyramid Scheme" Beast Boy goes into labor as a chicken and lays an egg.
Also, this isn't the first baby chick at Titan's Tower. In "Hose Water", Cyborg found a baby chick that he and Starfire bonded over.
Two of the carnies (which are villains) from "Fish Water" were seen running the party games at the carnival Beast Boy took his chicken son to attend.
The title of the episode may refer to the Harry Chapin folk song, "Cat's in the Cradle", which is about a father who isn't always around for his son.
Beast Boy's animal-related tomfoolery parallels gags and jokes seen in cartoons, such as alley cats keeping people awake at night, kangaroos boxing, octopuses drumming in an underwater band, and cows making milkshakes. Other gags that are referenced include Saint Bernard dogs wearing a barrel around their neck and alligators bushing their teeth with a toothbrush.
Superman's son, Jon Kent (Superboy), made an uncredited cameo in the episode on a title card screen that says "Father Son Bonding Time" right before the scene where Beast Boy goes to the carnival with his chicken son. It shows Jon playing football with his father Superman. This also marks the first known appearance of Jon Kent in animated media.
There is imagery of various DC characters at the carnival. Joker and Harley's faces are on the entrance sign, Trigon's face is on the Dunk a Dad booth, Red Tornado's head is on the Ferris wheel and Deadshot can be seen on the sign for the darts game.
The alley has graffiti of Batman, Catwoman, Joker, and Rorschach.
Beast Boy drinks a milkshake, made with fresh cow's milk, which he normally wouldn't drink as a vegan. But since the milk presumably came from his own cow body, he might not have technically been breaking any rules.
A title card screen titled "Fun With Parenting" shows Darkseid and baby Orion. Even though they are father and son, Orion was actually raised by Highfather Izaya from a very early age.
Beast Boy being too distracted by his animal shenanigans to bond with his chicken son.
The transcript for "Chicken in the Cradle" can be found here.
The image gallery for Chicken in the Cradle may be viewed here.