Bring 6 quarts water to boil, add 2 tablespoons salt.
Slice the trimmed artichokes up and down 1/4-inch thick to yield 4 to 5 pieces per artichoke. Place artichokes in large saute pan with garlic, onion, jalapenos and olive oil and saute over medium high heat until artichokes are tender but not too brown. Set aside.
Drop spaghetti into boiling water and cook until tender yet still al dente. Drain in colander over sink, reserving 1/2 cup pasta water.
Pour hot pasta into saute pan with artichokes. Turn heat to medium and gently toss to coat pasta with ingredients. If it seems too dry, add a little of the pasta water and toss again.
Toss in mint leaves, grated cheese and toss to mix. Pour into heated serving bowl, grind pepper over and serve.