I finally wrote up the free coaster pattern. Here it is. (Note: This picture is of the coaster before washing or blocking. The wonkyness settled down with blocking. I'd take a new picture but the coaster has since disappeared.)
Gauge 4 sts/inch in stockinette stitch (if you use cotton yarn then you need to remember that cotton shrinks and either wash your swatch or take that into account when measuring your gauge)
Needles size 7
Yarn worsted weight (I've used both cotton and acrylic yarns. Cotton has the added benefit of being absorbent)
(Use whatever yarn and needles you need to get the gauge above)
Finished size 4" x 4" (10 cm x 10 cm)
Cast on 36 stitches (I used longtail cast on. You can use whatever cast on you like, I don't think it really matters too much)
Row 1: K1,move the yarn to the front and slip next stitch purlwise,yarn back to the back, k1,move the yarn to the front and slip next stitch purlwise, k1... etc until you reach the last stitch. Slip it purlwise, making sure you move the yarn to the purl side so that it's in position to start the next row.
Repeat Row 1 until the coaster reaches 4" (10 cm) or until it reaches the point that the coaster is square
I'm not entirely positive this is the best way to bind off double knitting, but here's how I do it.
K2tog, k2tog, pass 1st st over second; repeat across. This makes for a very solid bind off, which I wouldn't use for something like a scarf or anything of that sort. But come on, it's a coaster. lol
This could easily be modified to having a different color on each side. Cast on the same way as the solid colored coaster. On the first row K1 with your MC, then after bringing both colors to the front purl the next stitch with CC. Making sure to keep both strands of yarn on either the front for purling or the back for knitting, work across alternating k stitches with MC and p stitches with CC. At the end of the row be sure to have both strands of yarn at the front so that the yarn is in the correct position to knit with it on the next row. You can even do colorwork patterns in this way just by switching around the colors. Experiment!