Preheat your oven to 400°F. Prick the eggplant with a fork and roast on a baking sheet for 30-40 minutes until the skin is charred and the flesh is very soft.
Once the eggplant is cool enough to handle, scoop out the flesh and discard the skin. Place the flesh in a food processor or blender.
Add the tahini, lemon juice, minced garlic, and a pinch of sea salt to the food processor. Pulse until the baba ganoush is creamy and smooth.
Season the chicken breast with half of the olive oil, smoked paprika, sea salt, and black pepper.
Heat a grill or grill pan over medium-high heat. Grill the chicken for 6-7 minutes per side, or until the internal temperature reaches 165°F.
While the chicken rests, dice the cherry tomatoes and cucumber. Toss them with the remaining olive oil and a pinch of salt.
Spread a generous portion of baba ganoush on a plate, top with the sliced grilled chicken, and serve the cucumber-tomato salad on the side.