Preheat oven to 350° F. 

Peel and halve mirlitons; simmer in a large pot of lightly salted water, covered, over medium heat until tender and easily pierced with a knife, about 10 minutes. Remove from water and rest upside down on paper towels to cool and drain. 

Meanwhile, heat 1 cup of oil in a large skillet over medium heat until warm. Add garlic; cook until soft. Stir in pepper flakes and bread crumbs; remove skillet from heat. Stir in Parmesan cheese, basil, and thyme. Set aside.

Heat remaining ¼ cup oil in another large skillet over medium-high heat until hot. Add crab meat, Sauce Ravigote, and green onions; sauté for 1 minute. Remove skillet from heat; stir in 3 tablespoons of bread crumb mixture. Season stuffing with salt and pepper. Set aside.

With a spoon, gently scoop out seeds in the middles of mirlitons to create a pocket for stuffing. Pack a spoonful of stuffing into center of each mirliton; place in a roasting pan or baking dish. Add remaining stuffing to mirlitons once they are in the dish. Cover with remaining bread crumb mixture. Bake stuffed mirlitons until stuffing is warmed through and golden brown, about 20 minutes. 

To make the Sauce Ravigote, combine all ingredients and refrigerate.