Preheat the oven to 350 degrees . In a food processor, puree the room-temperature butter, miso paste, chopped garlic and 2 tbsp. water; season with pepper. Starting at the neck end of the turkey, gently slide your fingers under the skin, separating the skin from the breast, thighs and legs. Spread the miso butter under the skin all over the turkey. Massage the outside of the turkey to help evenly distribute the miso butter. Season the cavity with salt and pepper. Place the turkey on a rack set inside a large roasting pan. Stuff the halved garlic head into the turkey cavity; loosely tie the legs together with kitchen twine. Brush the turkey with some of the melted butter; season with pepper. Pour 1 cup water into the roasting pan.
Roast the turkey, occasionally brushing with more melted butter, rotating the pan every half hour and tenting with foil if browning too quickly, and adding more water to the pan if needed, until an instant-read thermometer inserted into the thickest part of the thigh (not touching bone) registers 165 degrees , 2 1/2 to 3 hours. Let rest about 15 minutes. Carefully tilt the turkey so the juices from the cavity pour into the pan. Transfer the turkey to a platter. Let rest at least 15 minutes more.
In a large lidded saucepan, bring the stock and mushrooms to a boil over high heat. Remove from the heat, cover and let stand until the mushrooms soften, about 20 minutes. Using a slotted spoon, transfer the mushrooms to a cutting board; reserve the stock. Cut off the mushroom stems and discard. Thinly slice the caps.
In a small bowl, whisk 1/2 cup of the mushroom-infused stock and the flour until smooth. Using a large spoon, skim off some of the fat from the juices in the roasting pan; discard. Pour the remaining mushroom stock into the pan, leaving any grit behind. Place the roasting pan over two burners on the stove. Bring to a boil over medium-high heat, whisking and scraping up browned bits from the bottom of the pan. Strain into a large saucepan. Whisk in the flour mixture; add the chopped mushrooms. Simmer over medium, whisking often, until the gravy thickens, about 10 minutes; season.
Carve the turkey and serve with the gravy. Garnish with sliced scallions.
What's the Difference Between Red Miso and White Miso?