Step 1Halve cantaloupe and scoop out and discard seeds. Slice it into 1-inch wedges, remove skin and cut flesh into pieces about 1 1/2 inches long. Place pieces on a platter, uncovered, until needed.
Step 2Peel eggplant and cut it into 1-inch square pieces. Place pieces in bowl, liberally salting each layer. Cover with cold water. Place a plate over eggplant to hold it under water. Let it soak, to leach out any bitterness, 1 1/2 hours.
Step 3When eggplant has completed soaking, drain in colander. Rinse under cold running water to remove excess salt. Pat pieces dry with paper towels.
Step 4Pour 3/4 cup oil into medium nonstick skillet over medium heat. When oil is hot, saute eggplant until cooked yet firm, about 10 minutes, turning it with a spatula to prevent burning.
Step 5Remove eggplant with slotted spoon and place it on paper towels to absorb grease. Thoroughly clean skillet.
Step 6Mix together sugar and vinegar. Reserve until needed.
Step 7Return skillet to stove and add remaining 2 tablespoons oil. Turn heat to high and when oil is nearly smoking, add melon. Saute until it gains a bit of color and begins to expel its orange liquid, about 3 minutes. Reduce heat to low and add eggplant. Gently turn melon and eggplant with spatula. After about a minute, when eggplant is hot, pour in vinegar in an even stream. Turn again. Let vinegar bubble until its aroma rises, about 1 minute. Transfer caponata to serving platter to cool. Sprinkle with grindings of pepper. Serve at room temperature.