In a mixing bowl, combine the eggs, flour, baking powder, and olive oil. Whisk together while gradually adding the milk. Season with salt and pepper, and let the batter rest for 30 minutes.
Heat the olive oil in a small saucepan. Add the cornflour and cook for 2 minutes, stirring constantly. Gradually add the milk in two batches, stirring continuously until smooth. Season with salt and pepper, then set aside.
Dice the onion and courgette finely (brunoise). Sauté them in a lightly oiled frying pan for about 15 minutes until they are soft. Set aside.
Preheat and lightly oil a waffle iron. Pour the batter into the waffle iron and cook for about 4 minutes until golden brown.
Cut the waffles diagonally into halves. Spread béchamel sauce on each half, then top with the sautéed onion, courgette, and smoked trout. Close to form a croque-monsieur.
Just before serving, place the croque-waffles under a grill with a bit of béchamel sauce and grated Comté on top. Grill for about 4 minutes until the cheese is melted and bubbly. Serve with fresh basil.