Whisk together eggs, buttermilk, and butter, then stir into cornmeal mixture until just combined.
Cook ham in oil in skillet over medium heat, stirring, 3 minutes. Add jerk paste and cook, stirring, 1 minute. Transfer ham to batter with a slotted spoon and stir, then pour back into hot skillet. Bake until a wooden pick inserted into center comes out clean, 15 to 20 minutes. Cool in skillet 5 minutes, then invert onto a platter and serve warm or at room temperature.