Cut 1 Red Kuri squash in half and scoop out the seeds. Peel the squash carefully with a vegetable peeler. Cut into thin slices, then cut the slices into chunks.
Toss the squash on a baking sheet with 1 T. olive oil and 1/2 t. each black pepper and salt. Bake at 400 for half an hour or until the squash is just tender and browning on one side.
Cut the darker green leaves off of 1-3 leeks (depending on the size) and reserve for making broth, or discard. Slice the leeks in half lengthwise, then rinse under running water to remove any dirt that has collected.
Slice the leeks thinly to make 2 C. Saute on low heat in 2 T. olive oil over low heat with 1 T. fresh thyme or 1 t. dried thyme until completely soft.
Cut a cold 4 oz. log of goat chevre into chunks.
Using all the leeks, make a layer of leeks on the bottom of a 9 x 13″ baking dish. Arrange the squash above the leeks, along with chunks of the goat cheese.
Pour 1/3 C. milk over the entire gratin. Bake at 375 until the gratin is bubbling and the leeks on the bottom are nicely browned.