Preheat oven to 425°F.

 In an 8- or 9-inch cast-iron or oven-going skillet heat 2 tablespoons of the olive oil over medium heat.  Add the smashed garlic clove; cook 3 to 4 minutes or until golden brown, turning occasionally.  Remove the garlic; finely chop and set aside. Add the bread crumbs to the skillet; cook 4 minutes or until golden.  Transfer to a small bowl; stir in the chopped garlic, 1/4 teaspoon salt, and 1/8 teaspoon black pepper; let cool. Stir in the cheese and chopped parsley.

Cut the squash into four 1 3/4- to 2-inch- thick rings.  Using a spoon or melon baller, scoop out the seeds and flesh to create a 1 1/2- to a 2-inch-diameter hole.

Wipe the skillet.  Add the remaining 1 tablespoon of olive oil.  Add the squash rings, turning to coat all sides with oil.  Sprinkle with 1/4 teaspoon of salt. Bake for 15 minutes and flip the squash over.  Remove the thyme stems. Crack an egg into a small cup or bowl; slide it into a squash ring.  Repeat with the remaining eggs.  Sprinkle the eggs with 1/4 teaspoon salt and 1/8 teaspoon black pepper.  Bake for 10 minutes more or until the egg whites are set.  Top with the breadcrumb mixture. If desired, garnish with fresh thyme sprigs.  Serve immediately.