I was a children's dance teacher before I became a Mom and specialized in "Baby Ballerinas" as I called them...ages 3 thru 8. I thought I might be able to provide some insight :)
The reason you can't find a formal dance class for a two year old is because developmentally, two year olds aren't ready for a structured, formal dance class. Their attention spans aren't long enough yet and they don't yet have the self-control to handle the dynamics of a structured class.
It's one thing to LOVE spinning around in a tutu (what girl doesn't) but quite another to stand on one particular spot on the floor, and wait to do the movements your teacher tells you to do, or wait your turn in line to do 'across the floor' exercises while not bothering your neighbor and paying attention to the technique the teacher is asking you to follow.
I taught for a studio (for a very short time) that included 2 year olds in their beginning class and it was a nightmare...the girls were sweet, lovely girls but I must tell you that for a two year old, it is either torture or a joke to put new toys...I mean, tap shoes on their feet and then expect them to only tap when and how you tell them to. Frankly...even three year-olds aren't ready for tap shoes yet...tap movements yes! But the actual tap shoes shouldn't come in until their at least four.
You might be best suited to try out a "creative movement class" or another less formal opportunity to enjoy music, move to it and have fun with friends. If you start children too early in a class, when they're not yet ready, it won't be fun for them, they'll connect dance with being expected to hold still and/or follow directions (which they're not capable of doing but just because they're two!!! Not because they're not brilliant) instead of being able to find joy and freedom in beautiful movement and music! You want dance to be FUN! If you start too early then it won't be fun for them, or you...try dragging your daughter to a class when she doesn't want to go because it's more fun to twirl in the living room than listen to a teacher or wait her turn in line without playing with her friend's pretty tutu. Selfishly I'll throw out there that it's not that fun for the teacher to try to 'make' them ready before they really are...our job is to help your daughter find JOY in movement and music, but in order to do that she'll need to have some basic attention span and listening skills that(even some three year-olds)they don't have yet.
You'll be amazed at how much difference even 6 months can make! They'll go from just plain not being ready to suddenly have s BLAST!!!
A class for three year-olds (so you know what to look for when the time comes) will be tons of fun movements and silly, upbeat music. The teacher should be bubbly, energetic and enthusiastic. She should speak respectfully to the students and in my opinion...not be too 'baby talk' or condescending with them as they are not babies anymore and usually respond really well to reasonable expectations. You should expect the students to be taught proper terminology but in an age-appropriate way. They can learn the terms without it being too stuffy and it's surprising how proud they are of remembering the right term for a specific movement when you make it fun and positive.
Usually the class will be 45 minutes (3 and even 4 yrs is too early for a full hour) and generally will include 10 minutes of stretch/warm-up, 15 minutes of ballet (barre, across the floor and center floor), 10 minutes of tap (still in ballet shoes, just the basics of tap foot movements) and then 10 minutes of tumbling on mats or musical games, etc.
The class should flow from one section to the next smoothly and if the teacher is 'on her game' she'll keep a positive momentum and be in tune with the kiddos so when they start to lose attention, she's ready with a new activity or section to move along to. It makes it much easier on the kids to have a dynamic flow and not have big breaks where the teacher is trying to figure out what she wants to do next.
Lastly, I'm sorry to tell you but most studios don't even have classes on Fridays. Most families leave Fridays for weekend prep and if they go away for a trip or something, that Friday class is the first thing they skip so teachers and studio owners have learned from lack of attendance that it's not even worth it to schedule classes on those days. I know it's a bummer :( Sorry!!!
I hope this information was helpful and I welcome any further questions you may have :) Good luck and happy dancing!