Press the Fish. Slice the tuna and hamachi into roughly 1-inch thick pieces. Place each piece in a tortilla press and gently press until thin and flat, creating a fish sheet that will sit evenly on the tostada. Set aside while preparing the marinade.
Make the Quick Marinade. In a bowl, combine scallion bottoms, sliced jalapeños, soy sauce, fresh lime juice, and black pepper. Pour the mixture over the pressed fish and let it marinate for 2–4 minutes. This is a quick crudo-style marinade, not a long ceviche cure.
Fry the Tostadas. Heat neutral oil to 350°F (175°C). Fry corn tortillas until golden brown and crisp. Transfer to a rack set over a sheet tray to keep them crunchy, then immediately season with the tostada sour salt mixture for a bright, citrusy finish.
Make the Salsa Taquero. Add scallion tops, cilantro with stems, garlic, lime juice, olive oil, and salt to a blender. Blend until smooth and vibrant green, adding a little more olive oil if needed to keep the mixture moving. Transfer to a squeeze bottle and refrigerate until ready to use.
Prepare the Avocado Base. Thinly slice an avocado and gently fan the slices. Use a ring mold the same size as the tostada to cut a clean avocado round that will form the base layer.
Assemble the Tostadas. Layer each tostada with a crispy tostada shell, avocado round, pressed marinated tuna and hamachi, dollops of salsa taquero, extra scallions and jalapeños from the marinade, and fresh cilantro leaves. Serve immediately while the tostada is crisp.