Preheat oven to 450 degrees. Place diced eggplant in a large bowl and toss with 2 tbsp olive oil and a hefty sprinkle of salt. Arrange eggplant cubes on a baking sheet, evenly spaced apart and set aside. Add halved cherry tomatoes to the large bowl and toss with remaining 2 tbsp of olive oil and salt. Arrange cherry tomates, cut side facing upwards, on a baking sheet and set aside. Place both baking sheets in the oven and bake for 15 minutes then stir. Add balsamic vinegar to the eggplant. stir again, and place both baking sheets back in the oven for 10 minutes. Remove from oven, mix each sheet pan together and set aside.
Meanwhile, add pasta to boiling salted water and cook until al dente. Strain and set aside.
In a large sauce pan, add 5 tbsp of salted butter and melt over medium heat. Add garlic and a hefty pinch of salt and cook down for 2 minutes or until fragrant. Stir in heavy cream and add 1/2 tbsp salt and some fresh cracked pepper. Let simmer then slowly add in 1.5 cups parmesan, little by little, until melted.
Add cooked pasta to the parmesan cream sauce then top with roasted veggies, basil, more salt and pepper. Tear up two balls of burrata and stir them in. Season to taste.
Turn your oven on broil. Divide the pasta mixture evenly between four small cast iron skillets then top with the remaining parmesan cheese and butter. Broil on high for 2-5 minutes or until the parmesan is melted and slightly golden.
Before serving, halve the remaining two balls of burrata and place each half on top of each cast iron skillet. Finish with a drizzle of olive oil, a sprinkle of salt, some fresh cracked pepper and a few basil leaves. Serve immediately.