Bring a large pot of water to boil and add the kale leaves. Cook for a minute before transferring to an ice bath – this helps keep the sauce vibrant.
While that’s cooling, salt the same pot of boiling water and add your pasta.
Strain the leaves and squeeze out excess water, then add to a blender with 2 tbsp olive oil, a pinch of salt and 1/3 cup pasta water. Blend until smooth, then set aside.
In a sauté pan, heat the butter then add the anchovies. Sauté for a couple of minutes until the anchovies start breaking down, then use a whisk to break them apart until they’re homogenous with the butter – remove from the heat while your pasta finishes cooking so as not to burn the butter.
Once the pasta is just al dente, put the butter sauce back on a low heat and add the pasta directly to it, followed by the green sauce and a good amount of pasta water (1/4 cup). Stir, toss and agitate to marry and coat all of the pasta perfectly.
Divide between two bowls and serve topped with a dollop of burrata, crack of pepper and drizzle of olive oil.