Spray your donut pans with nonstick cooking spray. Beat together the eggs, brown sugar, milk, pineapple juice, sour cream, melted butter, & vanilla extract. Gently mix in the flour until just combined In the donut cavities, sprinkle about 1 scant tablespoon of brown sugar. Lay half a pineapple ring in each one (sliced in half carefully lengthwise so the cake/fruit ratio is even). Spoon the batter over the pineapple rings.