To the food processor bowl in which you've made the mayonnaise, add 2 tablespoons of the basil and a few drops of hot sauce. Pulse briefly until basil is minced. Taste and re-season if necessary. Set aside. (If making mayo by hand, mince the basil before adding with hot sauce.)
In a large bowl, toss grilled eggplant with a tablespoon of olive oil along with a generous pinch each of salt, white pepper, crushed red pepper, and the dried basil. Add the cooked potatoes, minced eggs, onions, bell pepper, celery, and the rest of the basil. Season with a little more salt (start with 1/2 teaspoon), pepper (1/4 teaspoon), and crushed red pepper (good pinch). Stir gently and add the white wine vinegar, 1 tablespoon of the lemon juice, and the reserved spicy basil homemade mayonnaise. Stir gently again until well combined. Taste and adjust seasonings, adding more vinegar or lemon juice if needed. Serve at room temperature or cover, chill, and serve cold garnished with a sprig of basil and the sliced egg. Store up to three days tightly covered in the fridge.
*I use the Daniel Boulud method (and add the basil/hot sauce) that uses a briefly poached egg, though for this salad I like 3/4 cup canola oil with 1/4 cup extra virgin olive oil rather than his 1 cup peanut oil.
**You can also sauté or oven-roast the eggplant, though the oiled cubes grill in just a few minutes on each side. To grill: Peel eggplant, chop into cubes, salt, let sit 10 minutes, pat dry before brushing with oil and grilling until there are dark grill marks on each side and the eggplant is tender.