Preheat oven to 400 F. Lightly butter a muffin tin, and set aside.
In a large bowl, mix together the olive oil, garlic, herbs, salt and pepper. Set aside for. now.
Using a mandoline, thinly slice the potatoes so they're about 1/8" thick. You can also attempt this with a knife, slicing the potatoes very thinly. However, it's easiest with a mandoline.
Toss the sliced potatoes in the oil and seasonings. Then add the grated cheese and grated broccoli, and toss well so that the potato slices are evenly coated.
Layer the potatoes evenly in the muffin tins. I ended up stacking 8 slices in each tin. It's ok if the potatoes are taller than the tin–they will shrink as they cook.
Tightly cover the muffin tin with foil and cook for about 20 minutes. Remove the foil and cook for an additional 10-15 minutes, until the potatoes are browned on top and on the sides.
Serve hot and crispy. Refrigerate leftovers and enjoy within three days.