Clean the anchovies: snap off each head and pull gently to draw out the spine and guts in one motion. Rinse under cold water and pat dry with paper towels.
Heat the olive oil in a 26 cm / 10-inch Dutch oven or wide pot over medium heat. Add the onion and cook for 5 to 6 minutes, stirring often, until soft and translucent.
Stir in the garlic and cook for 1 minute until fragrant. Add the tomato paste and stir for another minute to darken slightly.
Pour in the white wine and scrape up any browned bits from the bottom of the pot. Let it bubble for 2 to 3 minutes until reduced by about half.
Stir in the passata, stock (or water), red pepper flakes, salt, and black pepper. Bring to a gentle simmer.
Simmer the broth uncovered for 15 minutes, stirring occasionally, until it thickens slightly and turns a deep rusty orange.
Lower the anchovies into the broth in a single layer. Keep the heat low so the liquid barely simmers, no rolling boil, or the fish will shred apart.
Cook for 8 to 10 minutes, until the anchovy flesh turns opaque and flakes easily with a fork. Stir gently once or twice, breaking a few pieces into the broth.
Stir in half the parsley and taste the broth, adjusting salt and pepper as needed.
Toast the bread slices until deeply golden, then rub each one with the cut side of the halved garlic clove.
Set a slice of toasted bread in each bowl, ladle the bagnun over it, and top with the remaining parsley and a drizzle of olive oil. Serve right away.