Preheat oven to 375 degrees F and preheat a rimmed baking sheet in the oven.
Cut bread into 1/2-inch cubes. Combine 2 tablespoons of oil, 1/4 teaspoon of salt, thyme, and a pinch of black pepper in a large bowl. Add bread cubes and, using your hands, toss to coat. Transfer to a hot baking sheet, spread in a single layer, and bake until golden brown, about 15 minutes. If your oven has convection, turn it on to help brown croutons all over. Otherwise, turn them halfway through cooking for even browning. Remove and let cool on paper towels. Wipe the baking sheet clean, and then return it to the oven to get hot.
Rinse salmon and set on a sheet of heavy-duty foil.
Combine honey mustard, remaining 2 tablespoons of oil, dill, remaining 1/4 teaspoon of salt and remaining pepper in a small bowl. Spread over salmon and transfer to a hot baking sheet, using foil as handles.
Roast until fish is still filmy and slightly moist in the center, 10 to 15 minutes. Remove and let cool slightly.
Whisk together vinegar, sour cream, buttermilk, garlic, scallion, paprika, salt, and pepper in a large bowl. Whisk in oil in a slow, steady stream. Add romaine, greens, and carrots and toss gently to coat.
Divide salad among plates and scatter on tomatoes. Using tongs, break up fish into bite-size chunks and arrange over each salad. Top with croutons.