Dice 1 avocado and put in a small bowl with onion and jalapeño. In a saucepan, heat vinegar to simmering. Remove from heat; stir in 1 tbsp. kosher salt, the sugar, and cumin seeds until salt dissolves. Add 1/4 cup ice water and pour over avocado mixture, making sure ingredients are submerged. Chill, covered, 1 to 2 hours; then drain, transfer to a serving bowl, and chill until serving (up to 4 hours).
Meanwhile, in a food processor, purée flesh of 1/2 avocado with sour cream, 2 tbsp. water, 1/4 tsp. salt, and about 1/8 tsp. pepper. Transfer sauce to a small bowl and chill until serving (up to 4 hours).
Heat a grill to high (450° to 550°). On a work surface, carefully halve each chicken piece horizontally (keep the tenderloins attached). Place 1 breast half between 2 sheets of plastic wrap. Working from center out, pound with a mallet to 1/4 in. thick. Repeat with remaining chicken. Brush all over with some oil and sprinkle with about 1/2 tsp. salt and 1/4 tsp. pepper.
Thickly slice flesh of the remaining 1 1/2 avocados lengthwise. Brush with oil, sea-son with salt and pepper; set on a plate.
Grill chicken, covered, until grill marks appear, about 1 1/2 minutes. Turn and grill until meat is just cooked through, about 30 seconds. Transfer to a serving platter. Grill avocado slices cut-side down, covered, until grill marks appear, about 45 seconds. Flip avocados and repeat cooking. Transfer to platter.
Tuck tomato slices between chicken pieces on platter, and set out bowls of pickled avocado and sauce.