For the remoulade: Pulse the shallots and capers in a food processor until finely chopped. Add the mayonnaise, chives, mustard, smoked paprika, salt and pepper; pulse until well combined. Taste for salt and pepper and adjust. (The remoulade will keep, refrigerated, for 4 days. It is best made a day ahead.)
For the crab cakes: Melt 2 tablespoons of the butter in a saute pan over medium heat. Add the onions and peppers and saute until soft, 7 to 8 minutes. Add the garlic and cook until fragrant, 30 seconds. Let cool to room temperature.
Preheat the oven to 425 degrees F.
In a large mixing bowl, combine the eggs, chiles, crabmeat, 1 cup of the panko, mayonnaise, chives, parsley, salt, pepper and the cooled vegetables. Gently mix until well blended. Pack the crab mixture into a 1/4 cup measuring cup, then gently tap out into your hand and form into a patty. Place the patty on a baking sheet and repeat with the rest of the crab mixture. Spread the remaining 1 cup panko on a large plate. Dredge the crab cakes in panko on all sides.
Melt another 2 tablespoons of the butter in a large nonstick saute pan over medium-high heat. Add 2 tablespoons of the olive oil. When hot, add the crab cakes in batches, flattening them a bit with the back of a spatula. Cook until golden brown, 1 to 2 minutes per side. Between batches, add another tablespoon of butter and olive oil to the pan.
Place the browned crab cakes on a baking sheet and transfer to the oven for 8 to 10 minutes, until sizzling and heated through.
Serve the crab cakes with remoulade sauce on the side.