Preheat the oven to 400 degrees Fahrenheit. Set a rack in the middle position.
Peel most of the paper off of the garlic, leaving just enough to hold all of the cloves together and keep the head intact. Cut off and discard the top 1/4 of the garlic heads. Drizzle about 1 teaspoon of olive oil over each head and then wrap in foil. Bake for about 50-60 minutes, or until the garlic has become tender. Let cool for about 10 minutes and then squeeze the garlic out of the cloves and into a medium-sized bowl, and set aside.
In a medium or large-sized pan with a light colored bottom (so you can see the color of the butter as you cook it), heat the butter over medium heat. As the butter begins to melt, add the thyme leaves. The butter will begin to foam and crackle after about 30 seconds - give the butter a good stir at this point. As soon as the crackling noise subsides and the butter begins to turn brown, it will start to smell nutty, like toasted hazelnuts. Remove the butter immediately and pour it into a heatproof bowl and set aside.
Heat the oven to 425 degrees Fahrenheit and set the rack in the lower-middle position.
Wash the potatoes and pat them dry. Carefully cut slits in the potatoes, about 1/8-inch apart, making sure not to cut all the way through the potato, but almost so. Arrange the potatoes in a baking dish and brush them with half of the brown butter. Bake for 30 minutes.
Remove the potatoes from the oven and brush them with the remaining brown butter, making sure that some of the butter gets between the slices. Bake for another 20-25 minutes, or until the potatoes are almost tender but not quite done (the time may vary depending on the size of the potatoes).
Meanwhile, add the Gorgonzola and walnuts to the bowl containing the garlic and mix together until well blended. Set aside.
Remove the potatoes from the oven and spoon about 1 teaspoon of the Gorgonzola, walnut, and garlic mixture between every second or third potato slice. Sprinkle any remaining cheese mixture over the tops of the potatoes. Return the potatoes to the oven and continue to cook for another 10-15 minutes, or until the cheese is golden brown.
While the potatoes are finishing up, heat the pomegranate juice and balsamic vinegar in a medium or large-sized pan over medium heat. Bring it to a boil and then reduce the heat and bring to a simmer. Allow the juice and vinegar to simmer, stirring occasionally, for about 10-15 minutes, or until it has reduced by half and taken on a syrupy consistency (it should easily coat a spoon at this point). Pour it into a heatproof bowl and set aside.
Remove the potatoes from the oven and carefully transfer them out of the baking dish. Drizzle each potato with some of the pomegranate balsamic reduction and garnish with the pomegranate arils and a sprinkle of sea salt. Serve immediately.