Lightly spray two large rimmed baking sheets with cooking spray. On one sheet coat the eggplant and garlic cloves (skin-on) with 3 tablespoons olive oil and 1 teaspoon salt. On the second sheet coat the halved tomatoes with 1 tablespoon olive oil, balsamic vinegar, 3/4 teaspoon salt, and sugar. Put both baking sheets in a pre-heated 300° oven for roughly 1 hour or until the vegetables are shrivel-y on the outside and butter-y on the inside.
When the vegetables have about 20 minutes cook-time remaining, bring the water to a boil in a 6-8 qt. pot. Add 3 teaspoons kosher salt and pasta. Reduce the heat beneath the pot to medium to maintain a gentle boil, and cook the pasta until al dente (about 10-12 minutes). Once done, transfer the pasta to a large bowl using a slotted spoon or strainer and reserve at least 1 cup pasta water.
In a medium skillet, heat 1 teaspoon olive oil over medium heat. Add the turkey sausage and crumble it into small bits as it cooks through (about 8 minutes). Once done, pull the skillet from heat.
Remove and discard the skins from the roasted garlic cloves. Put the garlic cloves, roasted eggplant, 1/2 cup pasta water, 3/4 teaspoon salt, nutmeg, and black pepper in a large food processor. Puree until smooth. Add pureed eggplant mixture to the pasta and toss to coat adding more pasta water as needed. Turn the pasta onto a serving platter or up to 6 individual plates. Top with crumbled sausage, slow-roasted tomatoes, basil and grated Pecorino Romano cheese.