Lisa Henson and Emma Walton Hamilton met only recently, but they have something rare in common: each has a parent who likely held a deep, enchanted place in your childhood. Henson, who is fifty-six, is the oldest daughter of Jim Henson, the creator of the Muppets; she is now the C.E.O. of the Jim Henson Company. Walton Hamilton, who is fifty-four, is the oldest daughter of Julie Andrews, with whom she has written more than thirty children’s books.

“We met by Skype,” Walton Hamilton said the other day, sitting next to Henson in director’s chairs. They were at a sound stage on Long Island, on the set of “Julie’s Greenroom,” a Netflix series that premières on March 17th; both women are executive producers. The show, aimed at preschoolers, stars Andrews and a small band of puppets, who meet backstage at a theatre to learn about the performing arts. Each episode, Andrews welcomes a special guest to teach her Greenies, as she calls them, about a different discipline, including singing (Josh Groban), songwriting (Sara Bareilles), the makeup of an orchestra (Joshua Bell), and clowning (Bill Irwin). “We have a list of fifty topics we want to do, everything from hip-hop dancing to tap,” Henson said.

“Mime. African drumming,” Walton Hamilton added.

Onstage, Andrews, who wore a lavender sweater, sat in an armchair surrounded by puppet kids. The set was raised on a scaffold, with a breakaway floor to make room for puppeteers, who were arranged below in a Twister tangle. A sign overhead read “Are We Lucky, Or What!”—an Andrews motto. “It’s something that she has said for as long as I can remember,” Walton Hamilton said. “Sitting outside having a barbecue or something, she’ll go, ‘Are we lucky, or what?’ It can actually be under the worst circumstances—we’ll be in the middle of a thunderstorm and the power will go out.” (Hold the schnitzel with noodles.)

Today’s guest star: Alec Baldwin, teaching acting. The two performers rehearsed their scene. “Alec? You’re early, love!” Andrews said as Baldwin burst through the greenroom doors, wearing a gray T-shirt. She turned to her puppet gang and said warmly, “This is our stupendous guest and a very good friend, one of the world’s great actors, Mr. Alec Baldwin.” A puppet duck quacked in his face.

“No harm, no fowl,” Baldwin replied.

A few feet back, Henson and Walton Hamilton watched on a monitor. “Mom has a long-standing relationship with the Henson Company,” Walton Hamilton said, as anyone who remembers Andrews singing “The Lonely Goatherd” on “The Muppet Show” can attest. Walton Hamilton grew up in London, Switzerland, New York, and Los Angeles. Her father is the stage designer Tony Walton, Andrews’s husband before Blake Edwards; her godmother is Carol Burnett, another guest star on “Julie’s Greenroom.”

“I think your childhood was perhaps a little more cosmopolitan than mine,” Henson, who was brought up in Greenwich, Connecticut, said.

“It wasn’t very cosmopolitan,” Walton Hamilton countered, “because my mom was very protective of us and very careful to make sure that we had good bedtimes and cartoon breakfasts.”

“When I was in elementary school, if anybody found out that my father was a puppeteer they just felt sorry for me,” Henson recalled. “They were thinking church puppets.”

“But the difference is you carry the Henson name,” Walton Hamilton said. “I was able to hide a lot behind ‘Walton,’ and found that to be quite useful. People would look at me differently or expect things.”

“Did they expect you to be able to sing?”

“Exactly.”

Both women broke away from the family racket before rejoining the fold. Walton Hamilton moved to Sag Harbor in 1991 and founded a theatre with her husband, which they ran together for seventeen years. Henson, determined to prove herself “edgier than the Muppets,” became the first female president of the Harvard Lampoon, then moved to Hollywood and became the president of Columbia Pictures. “To be honest, I’m a little jealous of Julie and Emma getting to work together at this stage of life,” Henson said. (Jim Henson died in 1990.)

Henson had asked a company archivist to write up a time line of Andrews’s collaborations with her father. It resurrected memories for both daughters. 1973: The special “Julie on Sesame Street” airs. Walton Hamilton was ten. “I have a picture of me sitting on the step of a brownstone stoop with my mom and all the Muppets around us,” she recalled. “And Perry Como, for some reason.” 1975: The Muppets tape the special “Julie: My Favorite Things.” “The timing was such that we had to cancel a ski trip in Vermont,” Henson recalled. To make up for it, Andrews lent the Hensons her ski house in Gstaad, Switzerland. Lisa was fourteen.

“Let us know if you’d like to go skiing again,” Walton Hamilton told her. ♦

An earlier version of this story incorrectly stated that Jim Henson was a creator of “Sesame Street.” He did not create the show, but developed key characters for the series.