For the aioli: Whisk the egg yolk, lemon zest, lemon juice and garlic in a medium bowl. Combine the canola oil and olive oil in a spouted measuring cup. Fold a damp kitchen towel in half and place the bowl on top to anchor. Begin whisking in the oil, a few drops at a time, until the aioli is smooth and thick, increasing the stream when about a third of the oil has been incorporated. Season with salt and hot sauce and more lemon juice if needed. If the aioli is still very thick, whisk in a teaspoon or two of water. Refrigerate while you make the burgers.
For the burgers: Combine the beef and pork in a large bowl and sprinkle with salt and pepper. Form into 4 patties about 1-inch thick and 3 inches in diameter. Sprinkle the top side with salt.
Heat a large skillet over medium heat. Add the bacon and cook until crisp, 6 to 7 minutes. Drain on paper towels. Pour out all but 2 tablespoons of fat from the pan. Return the pan to medium heat and add the burgers, salted side down. Sprinkle the tops with salt and cook, flipping once, until completely cooked through, 4 to 5 minutes per side, adding a cheese slice to each in the last minute.
While the burgers cook, heat a large nonstick skillet over medium heat. Add the olive oil. When the oil is hot, break in the eggs. Sprinkle with salt and pepper. Cover the skillet and cook until the whites are set but the yolks are still runny, 2 to 3 minutes.
To serve, spread the aioli on the sourdough toast. Top each piece of toast with a burger, then a fried egg. Top each with 2 slices of bacon.