How to Make It
Make the sauce In a large saucepan, heat the olive oil. Add the onion, garlic and a generous pinch of salt. Cook over moderately low heat, stirring occasionally, until soft, about 20 minutes. Add the tomatoes, basil, thyme and bay leaf and cook over low heat, stirring occasionally, until thickened and reduced by half, about 3 hours. Discard the herb sprigs and bay leaf and pass the tomato sauce through a food mill into a heatproof bowl. Season the sauce with salt.
Meanwhile, make the béchamel In a small saucepan, combine the milk, garlic and thyme and bring just to a simmer over moderate heat. Remove from the heat and let stand for 10 minutes, then discard the garlic and thyme sprigs. In a medium saucepan, melt the butter. Add the flour and cook over moderate heat until a thick paste forms, 3 to 5 minutes. Very gradually whisk in the warm milk until smooth. Simmer over moderate heat, stirring, until thick and no floury taste remains, 5 to 7 minutes. Season the béchamel with salt.
Make the chicken In a large baking dish, whisk the flour with the cornstarch, garlic powder, oregano, cayenne and 1 1/2 teaspoons each of salt and pepper. In another large baking dish, beat the eggs. Spread the breadcrumbs in a third large baking dish. Season the chicken breasts all over with salt and pepper. Coat the chicken in the flour mixture, then dip in the beaten egg and dredge in the breadcrumbs, pressing to help them adhere. Transfer to a large baking sheet.
Preheat the oven to 425°. Line a large baking sheet with parchment paper. In a large saucepan, heat 3 inches of oil to 350°. Add 1 chicken breast to the saucepan and fry over moderately high heat, turning once, until an instant-read thermometer inserted in the thickest part registers 155°, about 7 minutes. Transfer to the baking sheet. Repeat with the remaining chicken breast.
Spread the béchamel over the chicken. Top each breast with 1 cup of the tomato sauce, 3/4 cup of the shredded mozzarella and 1/4 cup of the Parmigiano (reserve the remaining tomato sauce for another use). Bake for 5 minutes, until the cheese is melted. Transfer to a platter, garnish with basil and serve.