In a large saucepan, bring well-salted water to a boil and throw in the frozen artichoke hearts. Let them parboil for about 3 minutes, then run them under cold water and drain. Slice them into thin wedges. 

Slice the mushrooms or cut them into small pieces, depending on the size and shape of your mushrooms. 

Melt a knob of butter in some olive oil in a skillet or braiser over gentle heat. Add the minced shallots and let them soften well. Raise the heat to moderate and add the artichoke and mushroom, mixing them well with the shallots. Let everything sauté until both the mushrooms and artichokes pieces are almost tender.

Add a ladleful of the water in which the artichokes were parboiled or some broth, and let everything simmer until the liquid has reduced but not quite evaporated. Add some minced parsley if you like. Keep warm. 

Meanwhile, bring the water in which the artichokes have parboiled back to a boil. Add the tagliatelle. 

When the tagliatelle are still just slightly undercooked, add them to the skillet with the artichokes and mushrooms. Mix together, along with a good knob of butter, over very gentle heat until the tagliatelle have absorbed most, but not quite all, of the liquid. (Add more of the pasta water if need be to keep things moist). 

Serve immediately, with a sprinkling of minced parsley on top for color if you like.