Preheat the grill or a grill pan to medium high heat.
For the greens and croutons, brush the bread slices with the avocado oil and season with a pinch of salt and pepper. Cut romaine hearts lengthwise, leaving core to hold leaves together and place romaine hearts and bread on the hot grill for 1-2 minutes per side. Remove the greens to a platter and the bread to a board to cool.
For the dressing, in a medium mixing bowl combine the egg, Dijon mustard, anchovy, salt and garlic. Whisk to combine. Slowly begin to drizzle the avocado oil into the mixture, whisking constantly. The mixture will begin to thicken and emulsify with the oil. Continue whisking until all the oil has been incorporated. Add the juice of half a lemon along with the jalapeno seasoning. Taste the dressing and adjust with more lemon, salt and black pepper if desired.
Cut the toasted bread into cubes. Sprinkle the bread cubes and freshly grated parmesan evenly over the lettuce. Drizzle with dressing and serve.