In a small saucepan, melt the remaining 6 tablespoons of butter. Turn the heat up to medium and whisk constantly until the butter begins to brown. You can push the color a bit (more brown color equals more flavor) but be careful that it doesn't burn. When the butter is nice and brown, add the chicken stock and whisk together briefly. Allow the sauce to reduce down by about half or until it reaches a desirable sauce consistency, about 5 minutes. Season with a bit of salt to taste.
In a large bowl, toss together the cooked pasta, pine nuts, garlic, brown butter sauce and lemon juice. Stir in the parmesan cheese. Add some of the reserved pasta water as needed to help the ingredients combine.
Garnish with more grated parmesan cheese, if desired.
Notes
Replace the chicken stock with vegetable stock for a vegetarian recipe.