Bring a large pot of water to a boil. Drop in lobster tails and cook for 5-7 minutes. Remove and run under cold water or put in cold water to stop cooking and cool off. Remove meat from the shells and dice into small pieces.
At the same time cook the pasta until al dente according to the package instructions. Reserve a cup of the pasta water that you may need later.
Melt the butter over medium heat and then add the flour and whisk constantly. Cook your roux for about a minute to get the raw flour out. Add in about a cup of the milk and whisk to get all the lumps out before adding the rest. Add in the rosemary, thyme, cayenne, garlic, salt, and pepper. Increase the heat to medium-high and whisk continuously so that it doesn't burn as the sauce thickens. After about 5 minutes you should be good and have a thick sauce, béchamel.
Take the sauce off the heat and mix in the white cheddar and gouda a handful at a time. Mix well until the cheese is melted and cheese sauce is complete.
Preheat the oven to 400°. Add the pasta to the pot with the cheese sauce and mix together. If it looks too dry add some of the reserved pasta water. I added about 1/4 cup to make it a little more loose and creamier. Add the green onion and the lobster to the pasta and mix lightly to combine.
Add in half of the pasta to a buttered dish. Top with half of the mozzarella cheese. Add the rest of the pasta and then add the rest of the mozzarella to the top. Sprinkle the bread crumbs over the pasta. Put in the oven and bake for 25 minutes.