Warm the oil and butter in a large skillet over medium heat.
Toss the eggplant with the oil mixture and coat well.
Add salt and pepper to the eggplant, and cook until golden brown and very tender, 10-12 minutes. As the cooking process progresses, add more oil as necessary.
Season with salt and pepper and add the minced garlic, tomato paste, capers, and red pepper flakes.
The eggplant mixture needs to become uniform and loose, with some large chunks of eggplant still visible, and the flavors must have melded together (for another 5 minutes).
Take a taste of the eggplant mixture and set it aside.
Flour the work surface lightly before rolling out the dough.
It is best to roll the dough into a single layer using a rolling pin, a tabletop pasta roller, or a stand mixer attachment for making pasta.
Cut circles out of the dough with a 3- to 4-inch cookie cutter.
Portion out the filling across the top half of each circle with a piping bag or teaspoon. Keep it to a heaping teaspoon or so and avoid overfilling.
Wet the edges of each circle with a very thin swipe of water using a pastry brush or finger.
Use your fingers to seal the pasta into a half-moon shape by folding the bottom half of the round over the filling. If desired, use a pasta crimper or fork to lightly pattern the edges.
Boil a large pot of salted water.
Boil the mezzaluna for 2 to 3 minutes until tender and cooked through.
Transfer the cooked pasta to plates with a slotted spoon.
Over the pasta, spoon the warm eggplant mixture and serve immediately.