Preheat grill until very hot; toss in a handful of soaked hickory chips. When grill begins to smoke, in a large bowl toss mushrooms with 3 tablespoons oil and 1 tablespoon Creole spice or to taste; transfer to a wire grilling basket. Place basket on grill and stir mushrooms with a long pair of tongs for smoky flavor to penetrate evenly. Pour mushrooms onto a baking sheet to cool.

In a large pot of boiling water cook pasta until just tender; drain and keep warm. Meanwhile, heat remaining oil in a saute pan, add Tasso and saute, shaking pan frequently, 2 minutes. Add shallots, green onions and smoked mushrooms, and cook, tossing 2 minutes. Add garlic and cream, bring to a boil, reduce heat and simmer to reduce slightly. Add the butter to the sauce, adjust the seasoning with Creole spice and then add the pasta to the sauté pan and cook, tossing the pasta with the sauce until thoroughly coated and warmed through. Divide the pasta between 4 warmed plates. Garnish with green onions and Parmesan.