2 medium (or 1 large) Ciabatta loaves (for serving)
In a large deep saucepan over medium low heat, combine 1/4 cup oil and garlic. Cook, stirring occasionally, until garlic is golden and fragrant, 5 to 10 minutes. Stir in chile flakes; cook 30 seconds.
While garlic cooks, place tomatoes and juices in a large bowl and crush with clean hands. Remove tomatoes’ firm stem ends and any basil leaves packed into the can. Stir crushed tomatoes and salt into pot. Simmer over medium heat, stirring frequently, until tomatoes and garlic have completely broken down and sauce is thick, about 2 hours. Cool.
Line a large, rimmed baking sheet with parchment or paper towels. Trim stem end from eggplants; peel and discard skin. Using a knife or a Japanese mandolin, slice lengthwise into 3/16-inch-thick slabs. Arrange eggplant in a single layer on prepared baking sheet. Sprinkle both sides of eggplant slices with salt. Let stand 10 minutes. Pat eggplant dry with paper towels.
Line another rimmed baking sheet with paper towel and place a wire rack on top. In a large, deep skillet, heat 1/2 cup oil until a drop of water flicked into pan sizzles. Working in batches, fry eggplant until just tender, 20 to 30 seconds per side. Transfer fried eggplant to rack to drain. Remove skillet from heat.
In a large bowl, whisk together eggs, 2 tablespoons Parmigiano-Reggiano and 2 tablespoons pecorino. Add 3/4 cup oil to the skillet and return to medium-high heat until oil is sizzling.
Working in batches, dip drained eggplant into egg batter; fry in oil until lightly golden and cooked through, 2 to 4 minutes per batch. You will know oil is hot enough if batter puffs and sticks to eggplant on contact with oil. If it falls off, the oil isn’t hot enough.
Heat oven to 350 degrees. Spread 1/2 cup tomato sauce in the bottom of a 9-inch square baking pan. Arrange eggplant over the bottom of pan in a tight, even layer. Top with 1/3 of the remaining combined cheeses. Repeat layer of sauce, eggplant and 1/3 of cheese. Finish with a final layer of sauce and cheese (reserve any remaining sauce for serving). Transfer pan to oven and bake until cheese is melted and golden, 20 to 30 minutes. Remove from oven; Spoon additional sauce over top, and garnish with additional cheese. Cool for at least 20 minutes.
Cut eggplant into six equal pieces. Cut ciabatta into six pieces equal to the size of the eggplant servings. Split each portion of the ciabatta horizontally and toast. Sandwich eggplant portions between bread and serve.