To serve 6:
8 artichokes (they should be firm and feel solid -- soft or light artichokes will probably have fuzzy hearts)
The juice of a half a lemon
1/2 cup + 1 tablespoon olive oil for the artichokes
2 cloves garlic
Salt to taste
3 pounds baby potatoes (if they're small and fine-skinned they need not be peeled)
1 pint olive oil for the potatoes
A bunch parsley, minced
Pepper to taste
If the potatoes are young and fine skinned, wash and rub them with a rough cloth. Otherwise, peel them.

Trim the tough outer leaves off the artichokes (continue until the exposed leaves are almost all white), cut the tops off (perpendicular to the length of the artichoke) and cut them into eighths, putting the slices into water acidulated with lemon juice to keep them from turning black. When you have finished cutting them up, pat them dry and sauté them in a pan with the oil, garlic, salt, and minced parsley. Begin over a low flame, covered, and after a little while uncover them and turn them often so they cook well on all sides, browning and almost coming apart. When they're done drain away almost all the oil.
In the meantime heat the remaining oil in a high-sided pot suitable for frying, and add the potatoes in one fell swoop with a half cup of water. Let them cook gently at first, covering the pot so that they soften, and then raise the flame and uncover them to brown them.
Once the potatoes have browned, drain them and add them to the artichokes, together with salt and pepper to taste, and simmer for about ten minutes over a very low flame.