Whisk together the creme fraiche, mustard and hot sauce in a small bowl, and season with salt and pepper. Set aside.
Set a medium cast-iron skillet over medium-high heat. Add the bacon and cook until crisp, turning once, about 8 minutes. Drain on a paper-towel-lined plate, and then cut the slices into thirds.
Add the red onions to the skillet and cook, stirring frequently, until softened and starting to caramelize, 1 to 2 minutes. Stir in the mustard greens, sprinkle with salt and pepper, and cook until just starting to wilt, 2 to 3 minutes. Stir in the vinegar, and then transfer the greens to a bowl.
Whisk together the eggs, egg yolk and half-and-half in a baking dish until smooth. Add the Parmesan and some salt and pepper; whisk until combined.
Spread the bread slices with some of the creme fraiche mixture, and top with a slice of Gruyere. Top half of the slices with 3 or 4 pieces of bacon and a little mound of greens. Turn the remaining slices, cheese-side down, on top of the greens and press down to make sandwiches.
Working in batches of two, soak the sandwiches in the egg mixture on both sides until just soaked, about 2 minutes per side.
Heat 1 tablespoon of the canola oil and 1 tablespoon of the butter in a large nonstick skillet over medium-high heat. (If you have a skillet large enough to cook all four sandwiches at once, add all of the oil and butter now.) Add two sandwiches to the skillet, reduce the heat to medium-low, and fry until crisp, golden and cooked through on both sides, about 5 minutes per side--cover the pan if needed to help the cheese melt, and adjust the heat to prevent the French toast from burning. Repeat with the remaining oil, butter and sandwiches.
Halve the cooked sandwiches on the diagonal and serve hot.