Very cute pattern.
There are a couple of ways to do this but to stick closest to the pattern, cast on 8sts on the smaller needle with a new end of yarn. You can use a cable or knit cast on:http://www.knittinghelp.com/video/pl...ing-on-english
Then knit the 18sts from earflap 2 (which is on the larger needle), cast on 38 and knit earflap 1 etc. When you cast on the 38sts, you can just take the needle with the stitches on it in your left hand and do the cast on, then switch the needle back to your right hand.
There really is no reason not to break the yarn after earflap 2 since you'll be using a new end of yarn to start the cast on of 8sts onto the smaller needle and that yarn can be used with the rest of the join.