(Note: this isn't vegetarian, but will be if you substitute vegetable stock for chicken stock.)
Prepare the poblano peppers. Broil, turning every 5 min, until the skin is charred and the peppers are soft all over. Put in a paper bag and let sit for 10 min. Then skin under running water, also remove the seeds and stem. Dice.
Butter the bottom of a square pyrex dish. Use a mandoline to thin slice the potatos at 1/8th inch, or just slice very thin with a knife. Put the potatoes in layers, with some of the diced poblano pepper in between each layer. Also add a little salt and black pepper between each layer. Do four layers of potato.
Pour in the chicken broth. Mix the milk and cream and pour that in too. The liquid will bubble up as it bakes, so it shouldn't come right up to the top of the dish.
Preheat the oven to 400*F.
Cover tightly with tinfoil and bake until the potatoes are tender, about 1 hour. Then remove the tinfoil, sprinkle on the cheese, and bake until the liquid is thickened, about another 15 min.
So far I only made this once, but I liked it so much I wanted to write it down.
Preparing the poblanos is a decent bit of work, so next time I'll try using canned green chilis or maybe Mama Lil's. That said, poblanos do have a specific taste which I love.
This tasted very rich and cheesy, to the point where I plan to try it with less cheese on top next time.