Preheat your oven to 400 F (or fire up your grill for indirect cooking).

Cut off about 1/4" of the top of the garlic head and place on a piece of aluminum foil. Drizzle with the olive oil. Seal and bake for 45 minutes to an hour or until the garlic is tender and aromatic. Remove, open the foil, and let cool slightly. Squeeze the garlic from the cloves and mince.

Melt the butter in a large pot over high heat, but do not burn it.

Add in the flour and whisk for one minute. Reduce heat to medium.

Slowly pour in the milk while whisking. Simmer, but do not boil, stirring constantly until thickened.

Take a few tablespoons of the milk mixture and add to the egg. Beat gently with a fork. This will raise the temperature of the egg so that it does not curdle when you add it to the pot. Now, slowly add the egg to the pot.

Add in the cheese and stir until melted.

Stir in 2 tablespoons of the thyme, the red and black pepper, pasta and garlic, and season with salt. Remove from heat and let rest 5-10 minutes before serving.

Serve garnished with remaining thyme.

Note: If the pasta gets too thick, specially when reheating, just add a little bit of milk and stir.