Zest one lime and set the zest aside. Peel both limes, remove the white pith, and cut into segments over a bowl to catch the juices.
Trim and cut the salmon into 1/2-inch slices, then place the pieces into the bowl with the collected lime juice.
Add the sliced scallions, 2 teaspoons of the minced chives, 1 teaspoon of the minced mint, and 2 teaspoons of olive oil to the salmon. Season with salt, pepper, and Tabasco to taste, then cover and refrigerate while you prepare the other components.
Slice the tomatoes and combine them in a bowl with 2 teaspoons of chives, 1 teaspoon of mint, and 2 teaspoons of olive oil. Toss and season with salt and pepper; set aside until assembly.
Just before serving, cut the avocados in half, remove the pits, peel, and dice into 1/2-inch cubes. Place the cubes in a bowl, add the reserved lime zest, a squeeze of the remaining lime juice, the remaining 2 teaspoons chives and 1 teaspoon mint, a shake of Tabasco, and salt and pepper; gently toss to combine.
Gently fold the lime segments into the salmon and add additional lime juice or seasonings to taste.
Using four 4-inch ring molds, 1-cup ramekins, or glass serving dishes, layer one quarter of the avocado mixture into the bottom of each mold, then a quarter of the salmon, and finish with a quarter of the sliced tomatoes.
Drizzle a few drops of pistachio oil over each assembled tartare, then remove the rings to serve. If using ramekins without rings, line them with plastic wrap and invert to unmold.