In a large pot of generously salted boiling water, cook the pasta, stirring occasionally, until very al dente. Reserve one cup of the pasta cooking liquid before draining.

Meanwhile, in a large skillet over medium-low heat, melt the butter and combine with the oil. Add the garlic and cook, stirring frequently, for one to two minutes, until softened and fragrant. Increase the heat to medium, add the wine, tarragon, thyme, and simmer, stirring occasionally, for four to seven minutes, until you can no longer smell the alcohol and the liquid has slightly reduced. Stir in the lemon juice and red pepper flakes, if using.

Transfer the pasta and 1/2 cup of the reserved pasta cooking liquid to the skillet. Gradually add the Parmesan, a handful at a time, then the parsley, all while tossing; season with salt and pepper. Continue to cook, tossing frequently, for about two minutes, until all of the cheese is melted and heated through. Taste and season with more salt and black pepper. Add more pasta cooking liquid as needed, tossing, until a smooth, glossy sauce forms.

Divide the pasta among plates or bowls. Top with a poached egg, more grated Parmesan and a sprinkle of parsley before serving.