Heat the olive oil in a large heavy-bottomed skillet over medium heat.
Once hot, add the carrots, garlic, and a hefty pinch of kosher salt.
Cook for 4–6 minutes, stirring often, until the carrots are tender and the garlic is fragrant.
Add the ground beef along with another hefty pinch of kosher salt.
Break the meat into very small crumbles with a wooden spoon or spatula as it cooks.
Once the beef is browned and cooked through, check it with a meat thermometer for doneness.
Carefully drain or blot away any excess grease.
Stir in the marinara sauce and brown sugar.
Reduce the heat to low and simmer for about 10 minutes.
Meanwhile, turn on the broiler and arrange the buns cut-side up on a sheet pan.
Drizzle the cut sides generously with olive oil and sprinkle evenly with garlic salt.
Broil for 1–3 minutes, watching closely, until golden brown and crisp.
Spoon the Italian sloppy joe mixture onto the bottom buns.
Top each sandwich with torn burrata, flaky salt, freshly cracked black pepper, and sliced fresh basil.
Add the top bun and serve immediately while the burrata is perfectly creamy.