In a large skillet, melt butter over low heat. Add brown sugar. Stir constantly until melted. Carefully add pineapple rings. Simmer until caramelized. If desired, add 2 tablespoons pineapple juice from can to make a thinner glaze. Turn off heat. Set aside.
In a medium sized bowl, combine cornmeal mix, flour, corn muffin mix, 3 tablespoons of granulated sugar, eggs, and buttermilk. Stir together until smooth and pourable. Small lumps okay. Set aside.
Add 2 tablespoons vegetable oil to a 10" skillet. On top of stove over low heat, slowly preheat oil until it sizzles. Avoid smoking hot.
4. Again, stir batter until smooth. Carefully pour into hot skillet. Wear an oven mitt or use a thick towel to pick up HOT skillet's HOT handle.
5. Reduce oven to 375. Bake 20 to 25 minutes OR until knife inserted in center comes out clean and cornbread is golden brown. Watch carefully to prevent burning.
Drop 1/4 cup of cornbread batter into hot skillet. Spread thin with back of measuring cup. Cook until edges/top look dry.
Place a pineapple ring in center of flapjack. Place a Maraschino cherry in center of pineapple. Flip over. Cook another 2 to 3 minutes.
Use a rubber spatula to remove pineapple syrup from skillet. Drizzle over each flapjack before serving.