Combine goat's milk cheese and chives in a small bowl. Combine breadcrumbs, parmesan and basil in a shallow dish. Beat egg, egg white and salt in a second shallow dish.
Spread about 2 tsp goat's milk cheese mixture onto a slice of eggplant. Top with a tomato slice and an eggplant slice. Make 5 more sandwiches the same way.
Dip each sandwich in egg mixture, then breadcrumb mixture to coat both sides. Place on wax paper.
In large non-stick frying pan over medium-low heat, heat about 1½ tbsp oil. Add sandwiches in a single layer, working in batches, if necessary. Cook until fork-tender and golden brown, 10 to 12 minutes, turning over halfway through cooking. Add more oil as needed if pan becomes dry. Serve warm.