Preheat oven to 425 degrees . Cut about 1/3 inch off the top of the garlic head. Place on a 6" square of foil, drizzle with 1 tsp EVOO and sprinkle with salt and pepper. Wrap up the garlic in the foil. Place on a baking sheet; roast until fragrant and tender, about 45 minutes.
When cool enough to handle, unwrap the garlic and pull the cloves apart. Peel 3-4 cloves and mash into a paste (you'll need 1 packed tbsp).
Lower the oven temperature to 350 degrees . Spread the walnuts out on the same baking sheet; toast for 8-10 minutes. Let cool, then chop.
Using a food processor, blend the parsley until the chopped leaves stick to the sides of the bowl. Scrape down the sides, then add the walnuts, cheese and garlic paste; process to blend. With the machine on, slowly drizzle in the remaining 1/4 cup EVOO and the walnut oil. Season the pesto with salt and pepper.