Pulse the sugar and minced fresh rosemary in a food processor until moist and fragrant, about 30 seconds. Add tomatoes, pepper, and 1 and 1/2 tablespoons lemon juice to the food processor and pulse until fruit is finely chopped but not completely pureed and sugar is dissolved, about six 2-second pulses.

Pour the tomato mixture into a small nonstick skillet and bring to a boil over high heat. Cook over medium heat, stirring occasionally, until liquid has reduced noticeably, about 10 minutes. Adjust heat to medium and simmer, stirring more often as the mixture reduces, until it is glossy and has a jammy consistency between sauce and paste, 10 to 12 minutes.

Set aside off heat to cool to room temperature. Stir in remaining 1/2 teaspoon lemon juice. Taste and adjust seasoning with salt and pepper and additional rosemary, if desired.

Place a medium saucepan over medium high heat. Toss in the chopped bacon along with a big splash of water (a few tablespoons does the trick). Stir frequently with a wooden spoon until the water evaporates and bacon renders, releasing its fat. Continue stirring occasionally for 15-20 minutes until the bacon is evenly cooked. It should be nicely browned but not particularly crisp. Keep an eye on the heat, adjusting as needed so the bacon doesn't burn.

Remove the bacon from the pan and strain off all but 2 tablespoons or so of the drippings. Add in the onions and brown over medium low heat until they are soft and caramelized, about 30 minutes.

Add the bacon back to the pan along with the brown sugar and water. Bring to a boil and then simmer over medium heat until the water is absorbed and the mixture becomes thick and jam-like, about 20 minutes. Carefully transfer the mixture to the food processor and pulse until the jam is desired consistency.

Add four bacon slices to a large cold skillet. Turn the heat to medium high. Cook the bacon for 3 minutes, then flip the slices. Cook for 3 minutes on the other side and then flip the slices again. Turn the heat to medium if the oil starts to splatter and cook for an additional 2 minutes. The bacon should be evenly cooked and just a little crispy. Drain on a paper towel. Repeat with the remaining bacon slices.

Add one teaspoon of the bacon drippings to a small skillet set at medium heat. Gently crack the eggs into the skillet and cook for 3-4 minutes or until the whites are set. Flip the eggs and remove the skillet from heat.

Toast and butter the potato rolls. Add two slices of bacon to each roll, one egg and as much as tomato and bacon jam as desired. Enjoy!