Split eggplants lengthwise and score flesh deep but not through the skin. Salt generously and allow to rest for 5 minutes. Pat dry of excess water. Season with pepper and olive oil and grill flesh down first, then skin side until charred and tender.
Allow to cool slightly. Scoop out flesh and blend in food processor with olives, currants, basil and thyme, adding olive oil in a thin stream until emulsified. Mixture should lighter in color and thicker in texture. Taste, and add salt and pepper as needed.
Cut lavosh into desired shapes or sizes. Brush with olive oil. Season as desired with paprika, salt and pepper, and grill until crisp, 1-2 minutes per side.
Serve dip at room temperature or slightly chilled.