Heat the oven to 375°F. Toss bun cubes with melted butter on a sheet and bake for 8 to 10 minutes until crisp, turning once; set aside.
Shape beef into 4 thin patties and season with 1/2 teaspoon salt and pepper. Heat oil in a wide soup pot over medium-high heat and sear patties for 3 to 4 minutes per side until browned and 160°F in the centers. Transfer to a clean plate and cut into small pieces.
Leave 1 tablespoon fat in the pot and discard excess. Cook onion over medium heat for 6 minutes, then add cauliflower, broth and remaining salt. Simmer for 15 to 18 minutes until cauliflower is soft.
Lift 1 cup small cauliflower florets from the pot and reserve. Blend the remaining soup until smooth with an immersion blender, then stir in ketchup, mustard and milk.
Return beef and reserved florets and heat gently for 3 minutes. Turn off the heat and stir in cheddar in small handfuls until fully melted.
Divide among 6 bowls and top each with one-sixth of the toasted bun cubes and diced pickles immediately before serving.