Preheat a grill top or a grill to medium-low heat.
Place eggplants on the grill, and roast on all sides for 40 minutes, turning every 5 minutes. Immediately place eggplants on a plate, cover with plastic wrap, let cool for 15 minutes.
Remove eggplant stems, and peel off as much skin as possible. (It's okay if it doesn't all come off.)
In a food processor fitted with a chopping blade, pulse eggplant 7 times. Transfer eggplant to a medium bowl.
In a medium saucepan over low heat, heat 2 tablespoons extra-virgin olive oil. Add white onion, and saute, stirring occasionally, for 10 minutes. Add onions to eggplant.
Add garlic, lemon juice, salt, black pepper, red bell pepper, green bell pepper, and parsley to eggplant, and stir well.
Spread baba ganoush on a serving plate, and drizzle remaining 2 tablespoons extra-virgin olive oil over top. Sprinkle with cayenne, radishes, and green onions.
Serve cold or at room temperature.