It's this hat:
http://www.ravelry.com/patterns/library/daisy-cloche-3
So you do the brim in linen stitch, and with some short rows to get that shape, which I'm also having an issue with. At the end of a short row, she says to turn and "work one row even.", then do another short row, work one row even etc. But I think she means to the stitch marker, because otherwise the linen stitch pattern gets fucked up. Linen stitch is knit one, sl wyif. If I just keep knitting a full row even, after the short row I might have two knits or slips back to back and I'm not sure how to fix that. So I think maaaaaybe she means just knit back to the stitch marker and hopefully the pattern will line up again after I'm done with the short rows.
And you need the smaller needles because you keep binding off parts. After the linen stitch section is done, with size seven needles, you bind it all off, then pick the stitches up again with the size three, knit a few rows then switch back. You bind off again after the main body and I think after each section of the circles on the top of the hat. I assume this keeps the hat structured.