The beanie is worked in the round, in one piece, no seaming required.
One layer is knit in stockinette stitch; the other one is worked in lace pattern.

Size: one size (head circumference 54-56 cm).

You will need:
Yarn: sample is knit with Folco (100% Merino); it has 1500 m in 100 g, and I worked with 4 threads held together, which resulted in a fingering weight, ~375m/100g. I used ~120g. You will also need a scrap of waist yarn.
Needles: 3mm DPNs or circulars (in this case you will need 3 additional stitch markers to divide your work as it would be divided by DPNs).
Crochet hook, stitch marker, tapestry needle

Stitch patterns used:

Stockinette stitch (in the round): knit all stitches.
Lace stitch pattern: work from Chart 1.

Gauge: 28 sts. and 40 rounds in 10cm in stockinette stitch (after blocking).