In a medium bowl, combine 1 tablespoon lime juice with the chile, shallot and 1 teaspoon oil. With a rubber spatula, gently fold in the tomatoes and avocado. Set aside while you prepare the chicken.
Using a meat pounder, lightly flatten the chicken breasts so they're all the same thickness. 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 over-process or the nuts will form a paste. Transfer to a large plate. Add the cumin and 1/2 teaspoon salt and toss well.
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 remaining butter, oil and chicken. Season the tomato-avocado salad with salt and pepper, spoon it over the chicken and serve.