Preheat your oven to 325°F. Place nuts on a sheet pan and bake 5–10 minutes. The timing will be a little different for each, but as a rule of thumb, when the smell of toasted nuts starts to take over the kitchen, they’re done. Remove and set aside to cool.
Roughly chop the mint, basil, and spinach, and add to a food processor or blender.
Add the garlic, shallot, olive oil, nuts, lemon juice, ice cubes, and salt to the processor and blend on high until everything is incorporated. If you’re having trouble getting the pesto to blend, add a little more olive oil and a splash of cold water. Set aside.
Set a large pot of water to boil and salt like the ocean. Cook the pasta until al dente. If you’re using fresh pasta, this will happen fast, about 2–3 minutes.
In a large mixing bowl, off heat, mix the pasta with the pesto. Add a splash of the starchy pasta water and some olive oil to make the sauce creamy and delicious.
Plate the pasta into bowls and top with any extra bits of sauce, as well as a scoop of burrata, a few cranks of black pepper pepper, and a drizzle of olive oil.