Bring a medium pot of salted water to boil and poach the shrimp by placing the shrimp in the hot water, making sure it comes to a boil again, if it stops. Shrimp are done cooking when thoroughly pink, about 3 to 5 minutes or registering 145 degrees with an instant-read thermometer. Drain shrimp then transfer to a plate, place in the freezer or fridge until just chilled, about 10 to 20 minutes. Once cool, transfer to a cutting board and roughly chop and add to a large mixing bowl.
Meanwhile, heat the canola oil 1 tablespoon at a time in a medium to large skillet over medium-high heat. Once hot, fry the tortillas in batches, adding more of the canola oil, as needed until thoroughly crisp, turning them as needed, about 3 minutes per tortilla. Transfer to a paper towel-lined plate, season with salt. Repeat with remaining tortillas.
Add the cucumber, onion, jalapeno and lime and lemon juices to the shrimp mixing bowl. Mix well to combine and set aside.
When ready to serve, roughly chop the ⅓ cup basil and mix in with the shrimp ceviche. Slice open, seed and scoop out the avocados to a small mixing bowl. Season with salt and mash with a fork until you have a rough and chunky, yet spreadable consistency. Spread the avocado on the tostadas, top with the shrimp ceviche and tear more basil over the top.