For the casserole: Heat a charcoal grill to medium-high heat. Brush the eggplant with canola oil on both sides and season with salt and pepper. Grill until lightly golden brown and just cooked through, about 3 minutes per side.
Combine the peppers, pine nuts, garlic, and thyme in a food processor and process until coarsely chopped. With the motor running, slowly the cup olive oil and process until emulsified. Add 1/4 cup Parmesan and some salt and pepper and pulse a few times to incorporate.
Spoon about 1/2 cup of the pesto over the bottom of a baking dish. Line the dish with the some eggplant slices and then the mozzarella slices. Season with salt and pepper and scatter the oregano leaves over the top. Dollop another 1/2 cup or so of the pesto over, then more eggplant, a bit more pesto and finally the remaining Parmesan. Put on the grill, cover and bake until the cheese is melted and the casserole is hot, about 20 minutes.
For the breadcrumbs: Heat the oil in a medium saute pan over medium heat. Add the garlic and cook for 30 seconds. Add the paprika and cayenne and cook for 30 seconds. Add the breadcrumbs and cook, stirring occasionally, until the crumbs are lightly golden brown. Add the parsley and thyme and season with salt and pepper.
While the casserole cooks, make the breadcrumbs: Heat the olive oil in a heavy skillet over hot coals or over medium heat. Stir in the breadcrumbs (about 2 cups) and the spice rub and toast, stirring, until the crumbs are golden brown and crisp. Remove from the heat, stir in the parsley, and scatter over the casserole. Let rest 5 minutes before serving.