Preheat the oven to 375 degrees F.
Rub the cut sides of the squash with 1 teaspoon of the olive oil and sprinkle with salt and pepper. Set the squash cut-side down on a baking sheet. Cut the tops off of the garlic heads, exposing the cloves but leaving the heads intact. Set the heads on a large piece of foil, drizzle with the remaining 1 teaspoon of the olive oil and season with salt and pepper. Wrap up the foil to make a pouch and place on the baking sheet with the squash. Bake until the squash is tender and the garlic soft and roasted, 35 to 40 minutes. Set aside to cool.
When the roasted garlic is cool enough to handle, press out the cloves and add them to a food processor along with the minced garlic, toasted pecans, 1/2 bunch parsley and 1/2 teaspoon salt. Pulse until the ingredients are pulverized. With the motor running, drizzle in the 1/4 cup olive oil. Add the Asiago and lemon juice and pulse until just combined.
Shred the squash into a bowl using a fork and toss with 1/2 cup of the pesto. Season with salt and pepper. Serve garnished with chopped parsley.
Store any remaining pesto in the refrigerator for up to 1 week.