Preheat the oven to 150 degrees . Fill a large, deep skillet with enough salted water to reach a depth of 1 inch and bring to a boil. Add the asparagus and cook until crisp-tender, 3 to 4 minutes; drain. Reserve the skillet.
Heat the reserved skillet over medium heat. Add the turkey bacon and cook, turning, until browned, about 3 minutes. Transfer the bacon and the asparagus to an ovenproof dish and place in the oven to keep warm; reserve the skillet.
In a small saucepan, melt the butter over medium heat. Add the flour and cook, stirring, for 1 minute. Slowly whisk in the chicken broth and bring to a boil. Cook, whisking, for 1 minute. Remove from the heat, then whisk in the 2 egg yolks, 1 at a time. Stir in the lemon juice and season with salt and pepper. Strain the sauce into a bowl.
Fill the reserved skillet with enough water to reach a depth of 1 inch and bring to a boil. Crack the 4 whole eggs into the skillet and poach for 2 minutes for soft-set.
Place an English muffin half on each of 4 plates. Divide the bacon among the muffins and top each with asparagus, a poached egg and some sauce. Season with pepper.