In a blender, combine spinach leaves, dill, tarragon, basil, chives, garlic and lemon juice. With blender running, slowly add olive oil and blend until smooth and bright green. If you prefer it thinner, add a touch more olive oil. Season with salt and pepper. Add a little more lemon juice, if needed, to balance the herbs. Set aside. Sauce verte stays nicely in the refrigerator for up to 1 week.
Preheat oven to 325°F. Line a rimmed baking sheet with parchment paper. Set aside.
In a large cast-iron or nonstick skillet over medium heat, cook bacon, turning every 2 minutes, until bacon is browned and crisp. Remove and place on a paper towel to absorb fat and cool.
Pour off excess rendered fat, leaving a thin layer in the pan. With the pan still over medium heat, sauté 4 muffin halves in the remaining bacon fat, toasting on both sides. Place cut side up on the lined baking sheet and repeat with remaining 4 muffin halves.
Place a slice of cheddar cheese on each half. Place the baking sheet in the oven to melt cheese, 2 to 3 minutes. Remove from oven.
Crumble ½ slice bacon over each muffin half. Top with 2 slices of tomato and a light sprinkle of salt.
Crack eggs into a medium bowl and mix well with a fork to combine. Warm the skillet used earlier over medium heat and melt butter. Add eggs and season with a light sprinkle of salt and pepper. Using a spatula, gently push and fold the eggs, being careful not to overcook them. They should be soft and just set so they stay tender on the tartines.
Top muffin halves with equal amounts of eggs and a dollop of sauce verte. Enjoy!
