How to Make It
In a small saucepan, combine the vinegar and sugar and cook over moderate heat, stirring occasionally, until a medium amber caramel forms, 8 to 10 minutes. Carefully add the tomatoes and red pepper and cook, stirring occasionally, until the mixture is very thick, about 20 minutes. Transfer to a blender and puree. Strain the ketchup through a sieve. Stir in the ginger and season with salt and pepper.
In a large skillet, heat the 2 tablespoons of olive oil. Add the onions and cook over moderately low heat, stirring, until golden, 15 minutes. Stir in the sage and keep warm.
Preheat the oven to 475°. Heat a griddle. Lightly brush the griddle with oil and toast the focaccia on it until lightly golden, 1 to 2 minutes per side. Transfer the focaccia to plates.
Gently shape the ground beef into four 1/2-inch-thick patties and season with salt and pepper. Griddle the burgers over high heat until browned, 1 minute per side. Transfer to a baking sheet and top each one with a slice of cheese. Bake for 5 minutes for medium-rare. Set the burgers on the focaccia and top with the onions and a dollop of the ketchup. Close the burgers and serve.