Heat the oven to 425°F and arrange a rack in the middle.
Brush the cut side of the eggplant halves with olive oil and place them cut side down on a parchment paper lined baking sheet. Place garlic on a piece of parchment paper and pour remaining olive oil over top, fold up to enclose, and put on the baking sheet with the eggplant. Roast until eggplant skin remains indented when pressed on and the cut side of the eggplant is browned, about 35 minutes.
Set eggplant and garlic aside to cool, about 20 minutes.
Using a food processor fitted with a blade attachment, chop walnuts until they are reduced to the size of peppercorns, about 25 pulses. Add parsley and roasted garlic and pulse until parsley is evenly minced, about 10 pulses.
Scoop eggplant flesh from skin and add to the food processor along with lemon juice. Season with salt and freshly ground black pepper and blend until smooth, about 30 seconds. Serve with watermelon radish slices.
