Add the Pecorino-Romano pieces to a medium saucepan and cover with water (about 2 cups). Bring the water to a boil, then reduce the heat to medium-low and simmer for 30 minutes.
Meanwhile, add the garlic cloves to a small saucepan and add just enough water to cover. Bring the mixture to a boil, then turn off the heat and strain. Add the garlic cloves back to the same saucepan and repeat the process 2 more times. (This will bring out the natural sweetness of the garlic.) Allow to cool to the touch, then carefully shave very thinly on a mandoline.
Bring a large pot of salted water to a boil. Add the pasta and cook to very al dente according to the package directions.
Meanwhile, add the butter to a large saute pan and turn the heat on medium-high. Once the butter begins to melt, add the garlic and chile flakes. Let the butter melt and the garlic to fully bloom in the butter, about 1 minute. Strain the cheese stock directly into the saute pan, then stir and allow to come to a simmer. Add the cooked pasta and lemon juice to the sauce and finish cooking the pasta in the sauce. Add the chopped parsley and toss to combine.
Pour the pasta into a serving bowl and top with the arugula, then drizzle with olive oil and salt to taste. Top with the burrata. Gently open the burrata just a touch, then serve immediately.