1. In a 5- to 6-quart pan over high heat, bring about 3 quarts water to a boil. Add lasagna and cook just until tender to bite, 10 to 12 minutes. Drain and rinse with warm water.
2. Meanwhile, in an 11- to 12-inch frying pan over medium-high heat, frequently stir oil, onion, mushrooms, garlic, and basil until onion is limp, 6 to 8 minutes.
3. Mix broth and cornstarch. Pour into pan and stir until boiling. Add cream cheese and stir until melted.
4. Mix in sour cream, artichokes, and crab. Add salt and pepper to taste. Remove from heat.
5. Lay noodles flat. Divide all the crab mixture equally among noodles and spread the length of each strip. Roll noodles from a narrow end around filling. Set seam down, in a single layer, in a shallow 3- to 3 1/2-quart casserole. Cover with foil. If making ahead, chill up to 1 day.
6. Bake, covered, in a 350° oven until hot in center, 20 to 25 minutes (if chilled, 25 to 30 minutes). Uncover and sprinkle with jack cheese. Return to oven until cheese melts, about 5 minutes. Sprinkle with green onion.