Slice your zucchini in half lengthwise, and then slice them in very thin half-moon coins.
Set a medium pot of salted water on to boil.
In a large non-stick skillet, heat 2 teaspoons of oil until hot. Add the sliced zucchini and season with salt and pepper. Cook in an even layer, stirring now and again, until softened and starting to brown (about 4 minutes). Remove the zucchini to a plate.
To the same pan (don't wipe it out!), add another teaspoon of oil. Add your minced garlic and cook it until fragrant (about 30 seconds). Add the cherry tomatoes and capers and cook for a few minutes until the tomatoes start to release their juices (it helps to press down on them with a wooden spoon to speed this along).
To the pan, add the cooked zucchini and mint leaves (roughly chop them right before adding). At the same time, add your fresh pasta to the pot of boiling water. Cook the pasta for 2-3 minutes (until not quite al dente), while you are cooking the pan of vegetables for the same amount of time.
Take the pan of vegetables off the heat when the pasta is ready. Drain the pasta (reserve 1/2 cup of the pasta water before you drain it).
Add the cooked pasta to the pan of vegetables along with the 1/2 cup of pasta water. Season with salt and pepper. Cook for about 2 minutes. Remove from the heat and stir in the butter and parmesan cheese.
Serve topped with the fresh mozzarella cheese -- torn into bite-sized pieces.