In a large bowl, beat together the softened cream cheese, shredded cheese, drained tomatoes, ground cumin, cayenne, and lime juice until well combined.
Fold the shredded cooked chicken into the cheese mixture and season with kosher salt and freshly ground black pepper to taste. Cover and refrigerate until ready to fill tortillas.
Line a rimmed baking sheet with foil and place a metal rack on it; set near the stove for draining fried taquitos.
Pour oil into a Dutch oven or deep skillet to about 1/2-inch depth and heat to 325–350°F (use a thermometer if available).
Stack 6 corn tortillas on a microwave-safe plate and cover with a clean, damp cloth. Microwave 30 seconds on high, flip the stack, cover again, and microwave another 30 seconds to soften.
Remove one warm tortilla, place it flat, and spoon about 2 tablespoons of the chicken mixture 1 inch from the edge. Shape the filling into a tube and roll tightly from the edge closest to the filling, sealing the seam.
Using tongs, gently place the taquito seam-side down into the hot oil and hold the seam with the tongs until it stays closed. Fry until golden brown on one side, then flip and brown the other side.
Transfer the fried taquito to the prepared rack to drain. Repeat warming, filling, and frying with the remaining tortillas, working in batches.
Serve the taquitos hot with guacamole, salsa, lettuce, sour cream, cheese sauce, and sliced black olives as desired.