Prepare a grill fire to 350°.
Wash mushrooms and dry on a paper towel.
Toss mushrooms with olive oil and sprinkle with dry rub. Thread onto skewers.
Cook sausage links and mushrooms over indirect fire with the grill closed for 8 - 10 minutes.
Turn the sausages and mushrooms and continue to cook until sausage reaches 160° using a digital meat thermometer, about 15- 20 minutes total.
Remove sausages and mushrooms to a baking sheet. Tent with foil to keep warm while you cook the pasta and sauce.
Cook spaghetti in boiling water, drain well.
Meanwhile, make the sauce in a saucepan, adding heavy cream, parmesan, pesto and kosher salt. Whisk together and cook over medium heat. Whisk often until sauce is thickened and heated through.
Add the cooked spaghetti to the sauce and toss to coat.
Slice the sausage links and mushrooms and stir into the pasta.
Garnish with diced tomatoes and basil.