According to Reformed theology, what is an infant's first sin? From what I understand, every human that has ever lived is "born into sin", which means we all carry a "sin nature". Saul says:


  "For all have sinned, and come short of the glory of God" Romans 3:23


This means even little cute babies. If I'm not mistaken, most churches believe life begins at conception (which I have no problem with), so that means as soon as a baby is conceived, they are sinners before God. 

Just to clarify, I'm not looking for answers from those that hold to the "age of accountability". If anyone believes in this, please leave a comment and I'll ask a separate question.

My question is...

What is the first sin an infant (or fetus) commits? 

I've considered that this may be too broad of a question, but since infants can only do about 10 things (lay down, babble, drink milk, poop, etc), it shouldn't be hard to pinpoint that one thing that causes the infant to be a sinner deserving to have it's flesh burned forever and ever in agony.
Your hypothetical developing baby's first sin was disobeying God in eating the forbidden fruit.

On a more fundamental level, you are conflating two different concepts here. According to Reformed Theology (and indeed most branches of Protestantism) there are two different concepts in play here. Original sin, or the state in which we are born is different from specific acts of sin and the latter does not cause the former. Baby Jane doesn't have to do anything specific to be born sinful. She gets that from Adam—and by virtue of being a human.

The main doctrine you should read up on is called federal headship. This doctrine explains how all of mankind is considered fallen an account of Adam's sin. There are a number of good resources on the topic listed on Monergism and a summary on CARM.

Additionally the Reformed teaching of "salvation by grace alone" (sola gratia) is relevant here: salvation cannot be earned by by not sinning nor can it be lost by committing sinful acts. Believers can and do sin and don't automatically lose their salvation while non-believers may do "good" deeds that don't earn them favor from God because they are still inherently separated from him on account of Adam's sin and their fundamental nature. Baby Jane will do things that are selfish, defiant, and sinful starting even in the womb because that is the nature of a human. We do not wait around for our first acts of sin to be considered sinful. We deserve hell (separation from God) on account of our nature before we've done any specific acts of any moral importance.