Preheat the grill over medium-high heat and the oven to 275 degrees F.
For the eggplant puree: Place the Japanese eggplant slices into a perforated hotel pan over a solid hotel pan or on a rack set over a sheet pan and season with salt. Season the Italian eggplant flesh with salt and place into the hotel pan or on the rack with the Japanese eggplant slices, let sit about 5 minutes.
Place the eggplant into a bowl and add a drizzle of the extra-virgin olive oil, rosemary leaves, and thyme leaves. Place the whole eggplant onto the hottest part of the grill, flesh-side down. Cover with metal bowl and cook, about 5 to 6 minutes. Place the Japanese eggplant slices onto the cooler part of the grill and cook for about 2 to 4 minutes per side. Both the Japanese and Italian eggplants should be done at the same time.
Remove the charred flesh of the Italian eggplant and discard the skins. Add the flesh and seeds to a blender. Drizzle in about 1/2-cup extra-virgin olive oil, juice of 1 orange, and salt, to taste. Taste and re-season, if needed.
For the lamb: Season the lamb loins with kosher salt and freshly ground black pepper on both sides. Place a saute pan over medium-high heat and add the oil. Once heated, add the lamb to the pan and sear until golden brown, about 2 minutes. Flip and continue to sear on the other side, another minute. Add the butter, remaining rosemary sprig and thyme sprig and allow the butter to melt. Baste the lamb with the melted butter and place into the oven. Roast the lamb for about 10 minutes, or until the lamb is medium-rare to medium, about 125 to 130 degrees F.
When the lamb is done, remove the lamb from the oven and set aside briefly to rest prior to slicing. (2 servings)
For the sweet and sour relish: To a small saucepan over medium heat add 2 cups white wine and 1 cup golden raisins.
To a high-sided saute pan over medium heat, add 2 to 3 tablespoons blended oil. Once heated, add the shallots and season with salt, to taste. Next, add the garlic and stir to combine. Once fragrant, add the diced eggplant and season again with salt. Add the pine nuts, capers, the juice of 2 oranges, and about 1/2 cup sherry vinegar. Once the flames stops, add the raisins and wine and stir to combine. To a small saute pan, add about 1 tablespoon cumin seeds and place over low heat to toast. Once toasted add the cumin seeds and about 2 tablespoons sugar. Allow the eggplant to cook down and simmer for about 4 to 5 minutes.
For plating: To a medium bowl add the arugula, sliced chile and drizzle with extra-virgin olive oil. Toss to combine.
For each serving, spoon some of the smoky eggplant puree onto a plate. Slice half the tenderloin into 1/2-inch rounds and plate in between the grilled Japanese slices. Top with some of the sweet and sour eggplant relish and the micro arugula salad. Drizzle with extra-virgin olive oil.
Adding the salt to the eggplant will help draw out the moisture as it sits. When making the relish, after about 2 minutes, the alcohol will flame up, be careful. Allow the flame to die down.