Stir together the lime juice, lime zest, chili powder, cumin, sea salt, and black pepper in a medium mixing bowl to form a paste. Add the pieces of steak, use your hands to coat.
Heat a medium cast iron grill pan over medium-high heat for 3 minutes.
Grease with 1 tablespoon of avocado oil. Add the steak to the pan, and cook for 6 minutes, turning halfway through.
While the steak cooks, slice the avocados in half lengthwise and remove the pits. Remove the skin, and quarter the avocados. Brush the avocados with the remaining avocado oil on all sides.
Remove the steak from the pan, and set it aside on a plate tented with aluminum foil. Grill the avocado slices for 8 minutes, turning halfway through.
Serve hot alongside the steak bites and salsa.