Cut off and discard ends from 1 orange, then cut fruit crosswise into slices about 1/4 inch thick. Put in a 10-inch ovenproof nonstick frying pan (with curved sides).
To frying pan, add 1/4 cup water and 1/4 cup sugar and boil over high heat, shaking to mix fruit, until liquid is almost gone and oranges are slightly caramelized, 6 to 8 minutes. With a fork, push slices into a neat pattern in bottom of pan.
Meanwhile, grate 1 1/2 teaspoons peel from another orange. In a food processor, combine grated peel, 3/4 cup (3/8 lb.) butter or margarine, 1 cup sugar, 2 cups all-purpose flour, and 2 teaspoons baking powder. Whirl to mix, then add 2 large eggs and 1 teaspoon vanilla. Whirl until smoothly blended. Spoon thick batter in dollops over orange slices in pan.
Bake in a 350 oven until top is well browned and cake begins to pull away from pan sides, about 45 minutes (about 30 minutes in a convection oven).
Remove from oven and let stand about 10 minutes.
As cake bakes, cut peel and white membrane from all 3 of the remaining oranges and slice fruit crosswise; put fruit in a bowl.
Run a thin knife between pan sides and cake. Hold a flat plate over pan and invert to release cake.
Serve warm or cool in wedges with reserved peeled orange slices.
