Season the chicken with the coriander, the salt, and the pepper.
In a large skillet over medium-high heat, add the neutral-flavored oil.
Add the chicken breasts to the hot oil and cook, turning them occasionally, until the centers are no longer pink and the meat reaches an internal temperature of 165 degrees F, about 8 minutes per side.
Transfer the chicken breasts to a cutting board and allow them to cool for 10 minutes.
Cut the chicken breasts into bite-size pieces.
In a food processor or blender, add 1/2 cup of the cilantro, 2 tablespoons of the olive oil, 1 of the garlic cloves, and the juice of 1 of the limes and process until smooth.
Season the cilantro mixture with the salt and the pepper.
In a medium bowl, add the cooked chicken and the cilantro mixture and toss to coat and set it aside.
In another medium bowl, add 2 tablespoons of the vinegar, the mustard, the honey, and the 1 finely chopped garlic clove.
Gradually add 1/2 cup of the olive oil to the vinegar mixture, while whisking constantly, until it is smooth and emulsified into a vinaigrette.
Season the vinaigrette with the salt and the pepper.
In a food processor or blender, add the avocado, 1/4 cup of the cilantro, 1/4 cup of the sour cream, the remaining vinegar, the remaining whole garlic clove, and the remaining lime juice and mix, adding the water as needed to thin the sauce, until smooth.
Season the avocado cream sauce with the salt and the pepper.
In a large bowl, add the romaine, the cabbage, the onions, the spring mix, the remaining 1/4 cup of the cilantro, the scallions, and about 1/2 of the prepared mustard vinaigrette and toss until well coated.
Evenly distribute the tostadas among four serving plates.
Top each of the tostadas with an even amount of the chicken mixture, the black beans, the corn, the salsa, and the pico de gallo.
Evenly add the lettuce mixture to the tostadas.
Drizzle the salads evenly with the avocado cream sauce.
Top the salads with the remaining sour cream.
Serve with the remaining mustard vinaigrette on the side.