I wish I had a single easy answer about how this can be done. In fact, the methodological approach used will depend on what question you are asking of the texts, and what shards you are interested in. This will vary.
I think there are a few different kinds of shards we are talking about and several of these are bound up in any given text.
1. actual history about Jesus
2. actual history about early Christianity
3. memories about Jesus
4. memories about early Christianity
First I must tell you that I am not at all comfortable with recovering #1. Given the nature of memory (both individual and community) and the oral consciousness that dominated the ancient world and the kergymatic nature of our documents, what we can recover about the historical Jesus is highly problematic. I won't yet say impossible, but I approach that position in my own thinking about these things. I operate on a "probability" scale (no relation to the scientific use of this word) and a very strict method which begins with what I call the criterion of theological reuse. If the Christians are rewriting a tradition in a new theological direction, then I take whatever they are rewriting as earlier and more likely related to the historical Jesus. For instance, the virgin birth stories are rewriting the paternal genealogies and the fact that Jesus has a name that places his birth in Nazareth. So I am fairly certain that Jesus was born in Nazareth to Mary and Joseph his biological parents.My own work on the Gospel of Thomas has recovered an early kernel of sayings - a speech gospel from Jerusalem (=shard #2), dating 30-50 CE. I used a variety of methods to get at this, including literary criticism, historical criticism, social memory approaches, orality theories, and a new form of tradition criticism.
The recovery of the actual history of early Christianity is not as troubling to me, only because we have Paul, who really is our salvation here. We have here one of the only cases of autobiographical material - which still isn't actually what happened, but it is sure closer to it than what Acts reports (or Eusebius). So I think it possible to tease out this history a bit more reliably.
The third and fourth kinds of shards, well that is the really fun stuff in my opinion. This is the material that helps us reconstruct the various types of traditions that were developing and their opinions of themselves.
My work on the Gospel of Judas has recovered a form of Christian Gnosticism (=shard #2 and #4) in the mid-second century which was at odds with the apostolic churches and which criticized them for what they considered to be inept ritual activities (baptism and eucharist) and disfunctional theology based on infanticide (the killing of God's son) perpetuated by none other than the cursed Judas. It is clear from this study that the apostolic church was not the dominant church yet, that they began developing more sophisticated theology in response to these Gnostic criticisms, and that there was a very vicious struggle between the apostolic Christians and the Sethian Gnostic Christians with very serious and very nasty accusations being hurled by both sides. And the time? Only a couple of decades after the Pastorals are written! The Sethians believed that they were the true Christians and that all the others were ignorant and in their ignorance worshiped the creator of this world rather than the supreme God.
Think about how much more we can learn about early Christianity when we add to the apostolic memories all the memories of the Christianities that did not survive. Their memories give us a foil for the apostolic, so that we can see what perhaps they wished to vanquish or deny and what threatened them so much that they felt the need to adjust their own theology and practices in response.