Place cherry tomatoes, slivered almonds, garlic clove, and a pinch of crushed red pepper flakes in a food processor, then pulse. Add basil, mint, and kosher salt, then pulse a few more times, until the herbs are finely chopped. With the motor running, slowly stream in extra virgin olive oil until the mixture is creamy.
Transfer the sauce to a bowl, then stir in freshly grated Parmigiano Reggiano. Taste and adjust seasoning.
Bring a large pot of salted water to a rapid boil. Add pasta, then cook according to package directions until al dente. Before draining, reserve about the pasta water.
Place the pesto in a large skillet with deep sides set over medium heat. When the sauce is warm, add the drained pasta along with most of the reserved cooking water. Toss thoroughly until the pasta is evenly coated. Transfer to a large platter or individual bowls.
Top each bowl with a bit of hand-ripped fresh mozzarella cheese. Garnish with a drizzle of extra virgin olive oil, a pinch of flaky sea salt, freshly ground black pepper, and a few small basil leaves. Enjoy!