Preheat the oven to 375 degrees F (350 degrees if using a Pyrex or dark-finish pie plate). Grease a 10-inch pie plate.

In a medium bowl, combine the eggs, oil, vanilla, and sugar, beating with a wooden spoon until thick, about 1 minute. Add the flour, baking powder, cinnamon, allspice, and nutmeg, and salt. Beat vigorously until the batter has a crumbly consistency, about 1 minute. Fold in the apples and mix until the apples are evenly coated and the batter becomes moist and smooth. There will be lots of apples, held together by a relatively stiff batter.

Scrape the batter into the prepared pan and smooth into an even layer with a metal spatula. Sprinkle the top evenly with the sugar. Bake on the center rakc of the oven 30 to 35 minutes, until the center is firm when gently pressed. A cake tester inserted into the center should come out clean, the top and bottom of the cake should be deep brown, and the apples should be tender when pierced with a knife. Remove from the oven and cool at least 1 hour in the pan on a folded tea towel. Serve warm or room temperature, cut in wedges from the pie pan.