Step 1Heat the oven to 200 degrees. Using a mandoline, slice the Japanese eggplants lengthwise into paper-thin slices. Plunge the slices in boiling water for 20 to 30 seconds, then remove with a slotted spoon and place on paper towel.
Step 2Line a baking sheet with a silicone mat or parchment. Place the eggplant slices one by one on the mat and flatten out . Bake until crisp and dry, about 60 to 75 minutes. Store in a closed container for as long as several hours before serving.
Step 3Raise the oven heat to 350 degrees. Wash the globe eggplants and use a small knife to cut six slits in each. Insert the garlic halves into the slits. Drizzle with olive oil and one-fourth teaspoon salt. Wrap each eggplant tightly in aluminum foil. Bake for 50 to 60 minutes or until fork tender. Cool enough to handle.
Step 4Cut the eggplants in half, scrape out the insides and discard the garlic cloves. Puree the pulp in a food processor until smooth.
Step 5Measure out 1 1/2 cups of the puree. Reserve for the flan. Use the rest for making caviar: stir in one-fourth teaspoon salt, one-fourth teaspoon nutmeg and one-eighth teaspoon black pepper. Set aside.
Step 6In a blender, combine the 1 1/2 cups puree, the heavy cream, eggs and egg yolks, one-fourth teaspoon nutmeg, one-half teaspoon salt and one-eighth teaspoon black pepper. Blend until smooth.
Step 7Grease 6 (4-ounce) ramekins with the butter. Divide the eggplant-cream mixture among the ramekins. Put the ramekins in a large baking pan. Fill the pan with hot water until it comes halfway up the sides of ramekins. Cover the pan with aluminum foil and poke a dozen holes in it so that steam can escape. Bake until the mixture rises a little bit and barely jiggles when you shake it, about 30 to 35 minutes. Remove the pan from oven and let the ramekins cool in the water for 10 minutes.
Step 8To serve, run a clean knife around the edge of each ramekin and invert the flans onto individual serving plates. Warm a tablespoon and scoop out an egg-shaped portion of the eggplant caviar to place next to each flan. Arrange two eggplant chips standing in the eggplant caviar on each plate.