Wash the eggplants, halve them lengthwise, and brush with olive oil.
Preheat the grill to medium heat, then place the eggplants cut-side down on the grill.
Grill for 5-7 minutes per side, or until softened and slightly charred.
Scoop out the grilled eggplant flesh and place it in a bowl.
Add the minced garlic, smoked paprika, salt, pepper, lemon juice, and 2 tablespoons of BBQ sauce. Puree until smooth.
Slice the baguette or ciabatta and toast on the grill or in a skillet for 1-2 minutes.
Spread the eggplant dip on the bread slices, then top with grated cheddar cheese and fresh arugula.
Drizzle the remaining BBQ sauce over the sandwiches and serve warm.
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