SAUTÉ AROMATICS: Heat 1 tablespoon neutral oil in Dutch oven over medium-high heat. Add 1 large diced onion and 2 diced green bell peppers with pinch of salt; cook 5-6 minutes until softened. Add 3 cloves minced garlic and cook 30 seconds.
BROWN THE BEEF: Add 1 1/2 pounds ground beef, 1 1/2 teaspoons kosher salt, and 1/2 teaspoon black pepper. Brown thoroughly, breaking it up, 6-8 minutes. Drain excess fat if needed, leaving about 2 tablespoons in pot.
SINGER METHOD FOR MOUTHFEEL: Sprinkle 2 tablespoons all-purpose flour over beef and stir constantly for 60 seconds so flour coats meat and loses its raw edge.
BUILD THE SLOPPY JOE BASE: Stir in 2 tablespoons tomato paste and cook 1 minute. Add 3/4 cup ketchup, 2 tablespoons yellow mustard, 1 tablespoon plus 1 teaspoon Worcestershire sauce, 1 tablespoon brown sugar, 2 teaspoons apple cider vinegar, and 1/2 teaspoon smoked paprika; cook 1 minute, stirring to combine.
SIMMER: Add 4 cups low-sodium beef stock and 15 ounces crushed tomatoes. Bring to gentle simmer and cook 15 minutes, stirring occasionally.
RELISH, RESTRAINED: Stir in 1 tablespoon dill pickle relish and simmer 2 minutes. Taste, then add up to 1 tablespoon more only if you want extra dill flavor.
BRIOCHE BUN CROUTONS (CHEF METHOD): Lightly smear 6 brioche hamburger buns with 6 tablespoons garlic butter. Brown in skillet until caramel-toasty. Cool, then cube. Dry at 250°F for 30 minutes. For peak crunch, leave out uncovered overnight.
SERVE: Ladle soup into bowls. Top with generous handful of shredded cheddar (1 1/2 cups total for 6 bowls) and let it melt. Finish with bun croutons and optional chives or extra pickle relish.