Put the cubed potato in a saucepan with enough cold salted water to cover. Bring to a boil over high heat. As soon as the water boils, remove the potatoes, drain thoroughly and spread out on baking sheet to air dry.
Set a rack about 6 inches from the heat source and preheat the broiler.
Heat the olive oil in a 10-inch pan or skillet over medium-high heat. Add the artichoke hearts and cook, tossing, until they are lightly browned, about 5 minutes. Add the potatoes, then the shallots and the thyme. Adjust the heat to low and cook until thyme smells delicious and the shallots soften, about 2 minutes. The potatoes should not brown. Stir in the ham and toss for another minute, just until it frazzles. Scrape the vegetable-ham mixture into a bowl and wipe the pan clean with a wad of paper towels.
Heat the butter in the pan over medium-high heat until the butter stops foaming. Tilt the pan to make sure the bottom and sides are well coated. Pour in the eggs. Shake the pan while stirring the eggs with a wooden spoon or heat resistant spatula. Lift the edges of the tortilla as they set, letting the raw eggs run into the empty space. Continue until there is no longer any runny egg. Spread the vegetables and ham evenly over the soft eggs, season with salt and pepper, lower the heat, and sprinkle with the cheese.
Put the tortilla under the broiler and broil until golden brown and bubby, 3 to 4 minutes. Slide tortilla out of the skillet onto a large platter. Serve hot or at room temperature.