Boil a large pot of salted water and cook the pasta according to package instructions until al dente.
While pasta cooks, place the chopped beetroot and heavy cream into a high-speed blender and process until completely smooth and pink.
In a large skillet, heat olive oil over medium heat and sauté the minced garlic for approximately 1 minute until fragrant.
Lower the heat and pour the beetroot cream mixture into the skillet, stirring constantly for 2-3 minutes until heated through.
Whisk the grated Parmesan cheese into the sauce until fully melted and integrated.
Reserve 1/4 cup of pasta water, then drain the pasta and add it to the skillet with the sauce.
Toss the pasta thoroughly to coat, adding the reserved pasta water 1 tablespoon at a time if the sauce requires thinning.
Season with salt and pepper to taste.
Transfer to serving bowls and garnish with goat cheese crumbles and chopped chives.