Using a meat pounder, lightly flatten the chicken breasts. In a large shallow dish, combine the lime zest and 1 tablespoon of the lime juice with the tequila and garlic. Add the breasts and turn to coat with the marinade; set aside for 10 minutes.
In a food processor, finely grind the almonds; do not overprocess or the nuts will form a paste. Transfer to a large plate. Add the cumin and 1/2 teaspoon salt and toss well.
In a medium bowl, combine the remaining 1 tablespoon lime juice with the chile, shallot and 1 teaspoon olive oil. With a rubber spatula, gently fold in the tomatoes and avocado.
Remove the garlic slices from the chicken marinade. Pat the almond mixture on both sides of each breast half to coat thoroughly.
In a large nonstick skillet, melt 1 tablespoon of the butter in 1/2 tablespoon of the olive oil over high heat. Add 2 of the chicken breast halves, reduce the heat to moderate and cook until golden brown on the bottom, about 5 minutes. Turn the chicken and cook until browned on the second side and cooked through, 3 to 4 minutes longer. Transfer the chicken to plates and wipe out the skillet. Repeat with the remaining butter, oil and chicken. Season the tomato-avocado salad with salt and pepper, spoon it over the chicken and serve.