Preheat the oven to 400°F and set a baking sheet aside so the brioche buns are ready to toast.

Heat a large sauté pan over medium-high heat. Add the ground beef, diced onions, and green peppers. Cook for 6–8 minutes, breaking the beef into small crumbles as it browns. The mixture should be evenly cooked with no pink remaining and the vegetables softened.

Drain off the excess grease to keep the filling rich but not oily.

Return the pan to the heat and add the water, brown gravy mix, salt, and pepper. Stir until the gravy mix dissolves, then let it simmer for 3 minutes, stirring occasionally, until slightly thickened. If the mixture feels too thick, stir in 1–2 tablespoons of water.

Reduce the heat to low and stir once more to prevent sticking.

Arrange the brioche buns cut-side up on the baking sheet. Toast for 4–5 minutes, until lightly golden and crisp in the center. Keep an eye on them—brioche can brown quickly.

Remove the pan from the heat and stir in the shredded mozzarella and cheddar. Mix just until the cheese melts and the filling turns creamy.

Spoon the cheesy mixture onto the toasted buns and serve right away while hot and melty.