Preheat the oven to 375°F. Set the oven racks to the upper third and lower third positions. Line a large baking sheet with parchment paper.
Slice off the ends of the butternut squash and peel the skin using a vegetable peeler. Slice the butternut squash in half and remove the seeds. Dice the squash into ½-inch cubes. Toss the squash with about 1 tablespoon of olive oil. Spread the squash onto the baking sheet.
Slice off about ¼ inch from the top of the garlic bulb, exposing the cloves of garlic. If you sliced off large chunks of garlic and would like to use them, remove those garlic chunks from the papery skin and gently place them on top of the sliced bulb of garlic. Place the garlic bulb on top of a small sheet of foil and drizzle about ½ tablespoon of olive oil on top. Wrap the garlic up in foil and place it on a pie dish.
Roast the squash for about 30 to 35 minutes, until the squash can be easily pierced with a fork. Roast the garlic for about 45 minutes. Let the squash and garlic cool to room temperature.
Add the roasted butternut squash to the food processor. Squeeze out the roasted garlic from its papery skin and add to the food processor. Blend everything until smooth.
Spread the butter on one side of each slice of bread. If you find the butter is difficult to spread, microwave the butter for 10 to 15 seconds to soften it. The total amount of butter you'll use depends on the size of your bread slices.
Scoop up ¼ cup to ⅓ cup of the butternut squash and garlic mixture and spread it on the unbuttered side of a slice of bread. Top with ¼ cup of cheese and then ⅓ cup of baby spinach. Lay another piece of bread on top with the buttered side facing up. Repeat for the remaining sandwiches.
Heat up a large pan on medium heat and add 2 sandwiches to the pan. Grill both sides of the sandwich the bread is browned and the cheese has melted, about 3 to 4 minutes each side. Repeat for the remaining sandwiches.
If you have any of the butternut squash mixture left, you can save it for making grilled sandwiches on another day, or you can add it to a pasta salad or a soup.
If you are starting with butternut squash that is already roasted, you will need about 2 cups.