Heat 1/4 cup of the olive oil in a large pot over medium heat. Add the onions and garlic and season with salt and pepper. Cook, stirring occasionally, until softened and just starting to brown, about 6 minutes. Stir in 2 cups water and the stock and cook over high heat until the mixture is boiling.
Meanwhile, cut the broccoli crowns into small florets, then peel and slice the stems. Add the stems and two-thirds of the florets to the boiling water. Cover and simmer over medium heat, stirring occasionally, until the broccoli is very soft, 10 to 12 minutes.
While the broccoli cooks, spread the remaining broccoli florets on one half of a rimmed baking sheet. Drizzle the florets with 1 tablespoon of the oil and season with salt and pepper. Arrange the bread on the other side of the baking sheet and drizzle with olive oil. Bake, flipping the bread halfway through, until the broccoli florets and bread are lightly browned, about 10 minutes.
Meanwhile, make the pimento cheese. Stir the cheese, roasted pepper, mayonnaise and hot sauce in a bowl. Season with salt and pepper.
Stir the spinach into the soup until just wilted. Working in batches, puree the soup in a blender until very smooth (see Cook's Note). Stir in the vinegar and season with salt and pepper. Ladle the soup into bowls and top with the roasted broccoli. Spread the pimento cheese on the toasts. Garnish the soup and tartines with the snipped chives and pepper. Serve.
When blending hot liquid, first let it cool for 5 minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.
When blending hot liquid, first let it cool for five minutes or so, then transfer it to a blender, filling only halfway. Put the lid on, leaving one corner open. Cover the lid with a kitchen towel to catch splatters, and pulse until smooth.