Here are my notes about how I made this variation based on the main Entrechat knitting pattern, which is available here.
First a note about extra supplies: You will need a set of dpns of the same diameter as your main (circular) needle to work the sleeves in the round (unless you prefer to use the magic loop technique). You will also need a bit more yarn than indicated in the pattern since you are adding sleeve length. I would guesstimate that you will need anywhere from 50 to 150 yards extra depending on the size you are working and sleeve length you want to knit.
The beginning of the pattern can be followed as written until Page 3 of the pattern, where it says "Repeat Rows 3 and 4 for the raglan increases until you have the following stitch count."
You simply ignore the stitch counts that are immediately following this statement, as well as the next two sentences, instead working rows 3 and 4 until you get the stitch counts given at the bottom of page 3 (they are bolded).
Then, on the next WS, instead of binding off the sleeve stitches as indicated, you place them on hold using two pieces of scrap yarn (so, purl to the end of that WS, then slip both sets of sleeve sts onto holders, removing markers).
Then, you can follow the pattern as written to finish the body of the shrug.
Once you are done with the body, it's time to work the sleeves. Place the held sleeve stitches onto your dpns, making sure they are evenly distributed.
Then, joining new yarn at the underarm (leave a yarn tail about 8" or 20cm, which you will later use to thread a needle to close up any holes/gaps), you pick up and knit two stitches from the edge of the underarm area. Place a marker between those two stitches and then knit all the sleeve stitches. Then, joining in the round, knit your sleeve stitches in the round for approximately 1" or 2.5cm.
At that point, work a decrease round: Slm, k1, k2tog, knit until 3 sts remain, ssk, k1.
Continue working in the round in stockinette, working a decrease round every 2 inches or 5cm. When you are 1" shy of your desired sleeve length, alternate working a purled round and a knitted round to produce garter ridges. When you have three ridges, knit one more round and then bind off purlwise (this will produce your final garter ridge.)
Voila!! A sleeve! Do the same to work the second sleeve and you're all done!
Note: Sleeve length can be easily customized depending on what you need: short sleeves, elbow length, long sleeves. Simply measure the child's arm from the underarm to where your want the sleeve cuff to be. As a reference you can also consult the standard sleeve lengths provided by the Craft Yarn Council for babies and children.