Heat a grill pan to high.
Place a large pot of water on the stove to boil. Add salt and cook rigatoni to al dente.
Place porcini in a small pot with about 2 cups of water. Bring to a boil, drop to simmer, and reduce liquid by half, 10 minutes.
While pasta and porcini are working, pour about 1/4 cup extra-virgin olive oil into a small bowl. Use a pastry brush to coat the portobellos in extra-virgin olive oil and place them on hot grill. Cook until dark and tender, 7 to 8 minutes, turning once. Remove and let rest.
Heat a large deep skillet with 2 tablespoons extra-virgin olive oil, 2 turns of the pan, over medium-high heat. Add shiitakes and cook a few minutes then add garlic and cook a couple minutes longer. Add wine and reduce a minute. Halve and slice portobellos then add to shiitakes. Scatter the rosemary into the pan and turn to distribute the herb. Season the mushrooms with salt and pepper. Stir in tomatoes. Remove the porcini from steeping broth with a slotted spoon and chop. Stir mushrooms into sauce then add a few ladles of the steeping broth. Do not use the very last of it: any grit in the dried mushrooms will have settled to the bottom of the pot. Let flavors combine for a couple of minutes more.
Drain pasta and add to warm pot. Grate some Romano down over the pasta and add a few ladles of the sauce to the pasta. Toss the pasta and sauce to coat, then transfer to a serving platter or into individual bowls and top with remaining sauce. Pass more cheese at table for topping, and bread for mopping. Yum-o!
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