Whisk together the olive oil, lime juice or vinegar, garlic, paprika, cumin, oregano, salt, pepper, and chili powder until the spices are evenly distributed. The marinade should be slightly thick and fragrant, with no dry spice pockets remaining.

Add the chicken thighs and coat every piece well, turning them several times so the marinade reaches every surface. Cover and refrigerate for at least 1 hour, or overnight for deeper flavor and richer color.

Combine the cilantro, jalapeño, garlic, yogurt, olive oil, lime juice, and salt in a blender. Blend until completely smooth and creamy. The sauce should be thick enough to spoon over the chicken without running off immediately. Chill until serving.

Combine the tomatoes, red onion, cilantro, olive oil, Dijon mustard, and salt in a bowl. Toss gently until everything is lightly coated. Let the salad rest while the chicken cooks so the tomatoes release a little juice and the onions soften slightly.

Preheat the oven to 425°F (220°C). Arrange the chicken skin-side up on a parchment- or foil-lined baking sheet, leaving space between each piece for even browning. Roast for 35 to 40 minutes, until the skin is deeply golden and crisp and the thickest part reaches 165°F. Broil for 2 to 3 minutes at the end if you'd like extra crispy skin, watching closely to prevent the spices from burning.

Transfer the chicken to a serving platter and generously spoon the chilled aji verde over or alongside each piece. Add the tomato salad on the side and serve immediately while the chicken is hot and the skin is still crisp.