Position an oven rack in the lower third of the oven and preheat to 450 degrees F.
Heat a large skillet over medium-high heat. Add 2 tablespoons of the oil and heat another 30 seconds. Add the eggplant and salt and cook, stirring often, until the eggplant is golden brown and soft, about 12 minutes. Remove from the heat and stir in the marinera, basil and 1/4 cup of the Parmesan.
Lay one sheet of phyllo on a cutting board. Brush it gently with some of the remaining oil. Lay another sheet on top of that sheet and brush with oil. Do this 2 more times until you have 4 layers of phyllo, finishing the top layer with oil. Cut the sheet into 12 squares. Press one square (with all 4 phyllo layers) oil-side down into each cup of a 24-cup metal mini muffin tin. Repeat with the remaining phyllo.
Divide the eggplant mixture among the cups, about 1 teaspoon per cup. Divide the mozzarella over the top of the bites. Sprinkle with the remaining 2 tablespoons Parmesan. Bake on the lower rack until golden brown and bubbly, 13 to 15 minutes. Remove to a wire rack to cool slightly. Serve warm or at room temperature.