Melt the butter in a large saucepan over high heat.
Stir in the minced garlic and chopped onion, and sauté for approximately 2 minutes until the onion becomes nearly translucent.
Incorporate the diced capsicum and cook for another 2 minutes, allowing it to soften.
Add the ground beef to the pan. Cook while breaking it apart, until it transitions fully from red to brown.
Mix in the ketchup, crushed tomato, tomato paste, Worcestershire sauce, mustard, brown sugar, crumbled beef bouillon cube, and black pepper. Stir thoroughly to combine.
Lower the heat to medium-low (or low, depending on stove strength), then cover the pan. Let the mixture simmer for 30 minutes, stirring once or twice to prevent sticking. For a quicker version, simmer uncovered for 15 minutes.
Uncover the pot and stir vigorously for about a minute until the sauce thickens. The final mixture should be hearty, not overly saucy or runny. Taste and adjust salt if needed.
Toast the rolls and butter them if desired.
Generously heap the warm filling onto the bottom halves of the rolls. Immediately layer with cheese slices if using, allowing the residual heat to soften the cheese. Close with the top bun and serve promptly.