Make the hollandaise: in a blender combine 4 egg yolks, 2 tablespoons fresh lemon juice, 1/2 teaspoon cayenne, 1/4 teaspoon salt, and 1/4 teaspoon black pepper.
Melt 1 cup (2 sticks) unsalted butter in a small saucepan over medium heat.
With the blender running, slowly pour the melted butter through the lid vent until the sauce is thick and smooth, about 1 to 1 1/2 minutes; thin with a little water if necessary and adjust seasoning to taste.
Preheat the broiler to high. Arrange tomato slices on a baking sheet, drizzle with 1 tablespoon olive oil, sprinkle 1 teaspoon dried thyme and salt and pepper to taste, then broil until softened and lightly charred, 1 to 3 minutes; remove and set aside.
Place English muffin halves on a baking sheet and broil about 1 minute until lightly crisp; set aside.
In a large frying pan, cook the breakfast sausage patties until lightly browned and cooked through, about 3 minutes per side; keep warm.
Bring a large pot of water to a gentle boil. Add 1 to 2 teaspoons distilled white vinegar to the water and create a swirling vortex with a whisk or spoon.
Crack 1 or 2 eggs into ramekins, then gently lower them into the simmering water (reduce heat to medium) and cook for 3 to 4 minutes until the whites are set but yolks remain soft.
Lift poached eggs with a slotted spoon and transfer to a bowl of warm water to stop cooking; repeat with remaining eggs and keep warm.
To assemble, place two muffin halves (cut side up) on each plate. Top each half with one or two sausage patties and a roasted tomato slice, then place a poached egg on top.
Spoon about 3 tablespoons of hollandaise over each assembled half and finish with a sprinkle of black pepper. Serve immediately.
