Preheat the oven to 300 F. Heat a large skillet over medium-high heat.
While the pan is preheating, cut off the stem of each eggplant then peel and discard the skin. Use a sharp knife to slice each eggplant in half lengthwise then cut each half into quarters or thirds depending on the size.
Pour about 1/4 cup of olive oil into the pan, enough to coat the surface in a thin even layer. Next, add half of the eggplant pieces to the pan (or as many as will comfortably fit without overcrowding). Cook until golden brown on all sides, flipping halfway through, about 4 minutes per side (you may need to adjust the heat between medium and medium-high as you go). Transfer to a plate and season with salt. Repeat with the remaining eggplant, using more olive oil as needed. Remove skillet from heat.
If your skillet is large enough to hold all of the eggplant, tomatoes and peppers then go ahead and use that. If not, a baking tray or high-sided casserole dish will work just fine. Place the seared eggplant, tomatoes, garlic cloves, peppers and shallots into your chosen baking vessel. Drizzle the olive oil over top and season generously with salt.
Transfer the mixture to the oven and bake for 1hr-90 minutes. Stirring every 30 minutes or so. It's ready once the vegetables have become completely soft and slightly caramelized around the edges.
Remove the skillet from the oven and fold in the basil leaves. Let stand at room temperature for 15 minutes. During this time the basil will "steep" into the sauce. While the basil is steeping boil off the pasta, reserve 1-cup of pasta water before draining.
Transfer the eggplant mixture to a blender or food processor. Pulse until all of the ingredients are finely minced but not completely smooth. Taste and season with additional salt as needed.
Add the sauce back into the skillet along with the cooked pasta and 1/2 cup of pasta water. Fold everything together over medium-low heat. Just before serving, crumble in the goat cheese. Divide between bowls and top with torn basil leaves. Serve warm.