Prepare a grill fire to 350°.
Using kitchen shears, cut the lobster tails in half lengthwise, through the shell to make 2 piece out of each tail.
Brush the lobster flesh with olive oil and season with kosher salt and pepper.
Grill lobster tails, flesh side down for 5 minutes with the grill closed. Turn the lobster tails and continue to cook until internal temperature reaches 145°.
Remove from grill. Tent with foil.
Melt 1/4 cup butter in a small skillet with 4 cloves minced garlic. Cook over low heat, stirring until garlic is lightly toasted. Remove from heat and cool 5 minutes. Stir in ground rosemary, dried thyme, smoked paprika, pepper and maple syrup.
Brush garlic butter over lobster tails and keep warm until pasta is ready.
Cook pasta in boiling water.
Meanwhile, make the lemon garlic sauce in a large skillet . Add 1/2 cup butter, 1/4 cup olive oil and 2 minced garlic cloves to the skillet. Cook over medium heat, stirring often for 2 minutes. Add lemon juice and a scoop of the pasta water. Whisk in lemon zest, salt and pepper. Cook 2 minutes.
Use tongs to remove cooked pasta from the water into the sauce. Add the parmesan cheese and toss until thoroughly combined.
Garnish with parsley and top with garlic butter lobster tails.