Lightly oil an 8-by-11-inch glass baking dish. In a medium heatproof bowl, mix the cornstarch with 1/2 cup of the sugar. Slowly whisk in 1 cup of the milk. Add the egg yolks and 1 of the whole eggs and whisk until smooth. Whisk in 1/4 teaspoon of the cinnamon.
In a medium saucepan, heat 3 cups of the milk over moderate heat until small bubbles begin to form around the edge of the surface. Gradually whisk the hot milk into the egg mixture. Return the mixture to the saucepan and cook over moderate heat, whisking constantly, until the consistency is thicker than mayonnaise, about 5 minutes. Stir in the vanilla.
Scrape the custard into the prepared baking dish and smooth the surface; let cool slightly. Refrigerate until very firm, at least 3 hours or overnight.
In a small bowl, mix the remaining 1/2 cup of sugar with the remaining 2 tablespoons of cinnamon. In a pie plate, whisk the remaining 2 eggs with the remaining 2 tablespoons of milk. Spread the flour in another pie plate.
Cut the custard into 24 rectangles. In a large, deep nonstick skillet, heat 1/2 inch of vegetable oil until shimmering. Working with 6 pieces of custard at a time and keeping the rest refrigerated, dip the custard rectangles in the beaten eggs, then dredge in the flour. Fry the custard over moderate heat, turning once or twice, until golden and crisp, about 5 minutes. Transfer to a rack lined with paper towels and immediately sift cinnamon sugar over them. Repeat with the remaining pieces of custard. Serve warm or at room temperature.
When making custards with vanilla or other extracts, stir the flavorings in after cooking because heat can dull their taste.