Heat a griddle pan or cast-iron skillet over medium-high heat.
Season the chicken with salt and pepper. Combine remaining spices in a small bowl and rub all over chicken. Drizzle griddle or skillet with a tablespoon of EVOO. Cook chicken 12 minutes, turning once. Remove from pan and let cool enough to handle; slice each piece on an angle into 4 pieces.
Halve and pit avocados and dress with lime or lemon juice. Cut each half into 4 pieces and layer between chicken slices.
In a small skillet, heat remaining tablespoon EVOO, a turn of the pan. Add onion, jalapeño and garlic, and sauté a few minutes to soften. Add tomatoes and season with salt and pepper. Turn off heat and mix in cilantro.