Step 1

To make the lemon-basil sour cream: In a small bowl, whisk mayonnaise, sour cream, lemon juice, basil, chives, and lemon zest. Season with salt and pepper to taste. Sauce can be made 8 hours ahead; just cover and chill until use.

Step 2

To make the chile oil: Add all ingredients to a blender. Blend, starting on low and gradually increasing to high speed, until completely smooth and the sides of the blender jar are warm to the touch, 2 to 5 minutes. Chile oil can be made 1 week ahead; store in an airtight container at room temperature until you're ready to use it.

Step 3

Heat a large skillet or griddle on medium-high and toast bread halves on both sides until deep golden-brown and warmed through, about 3 minutes per side. Set aside until ready to use.

Step 4

Place flour in a shallow bowl (a pie plate works best). Place eggs in another shallow bowl. Place panko in a third shallow bowl. Season flour, eggs, and panko with salt and pepper.

Step 5

Working one at a time, season tomato slices with salt and pepper, then dredge cut sides in flour, shaking off excess. Transfer to bowl with egg and turn to coat. Lift from bowl, letting excess drip off. Coat in panko, pressing to adhere, then transfer to a rimmed baking sheet.

Step 6

Heat 3 tablespoons oil (plus more as needed) in same large skillet over medium-high. Working in batches, Cook tomatoes until coating is deep golden brown about 3 minutes per side. Transfer to a plate. Wipe out skillet and repeat with remaining oil and tomatoes.

Step 7

Spread a few tablespoons of the lemon-basil cream on each slice of bread. Layer a few slices of cucumber over 4 of the pieces of bread, then top with 2 to 3 pieces of fried tomato, basil leaves, chives, Parmesan cheese, and a squeeze of lemon. Drizzle with chile oil and close the sandwich with the remaining slices of bread. Serve remaining cream alongside.