Place a rack in middle of oven; preheat to 350°. Heat 2 Tbsp. extra-virgin olive oil in a large saucepan or small Dutch oven over medium-low. Add 1 medium onion, chopped, and season with kosher salt. Cook, stirring occasionally and reducing heat if onion starts to brown, until softened but without taking on any color, 5–7 minutes. Add 3 garlic cloves, finely chopped, and cook, stirring, until softened and fragrant, about 2 minutes. Add 1 large head of cauliflower (21/2–3 lb.), florets broken into small pieces, core chopped, 11/2 tsp. Diamond Crystal or 3/4 tsp. Morton kosher salt, 1/4 tsp. freshly ground white or black pepper, and 51/2 cups water (the water should just barely cover the cauliflower). Bring to a boil, then reduce heat, and simmer, uncovered, until cauliflower is completely tender and starting to fall apart, 15–20 minutes. Stir in 3/4 cup half-and-half and simmer 5 minutes; remove from heat.
While the soup is simmering, whisk 1 Tbsp. Dijon mustard, 1 Tbsp. unsalted butter, melted, and 1 Tbsp. extra-virgin olive oil in a medium bowl to combine; season with salt and pepper. Add 5 cups country-style bread, preferably day-old, torn or sliced into 1" cubes, and toss to coat in mustard mixture. Scatter 1/2 cup finely grated Gruyère or cheddar over and toss again to combine. Transfer bread mixture to a parchment-lined rimmed baking sheet and bake in a single layer, turning halfway through, until cheese is melted and croutons are golden brown and crisp, 10–15 minutes.
Working in batches, carefully purée soup in a blender until very smooth, transferring to a medium bowl as you go. (Alternatively, you can use an immersion blender to blend soup in pot until smooth.) Return soup to pot and reheat over medium-low, stirring and adding more water to thin if needed (you’re going for the consistency of heavy cream). Stir in 1 Tbsp. plus 11/2 tsp. fresh lemon juice. Taste and season soup with more salt and/or pepper if desired.
Ladle soup into bowls. Top with cheesy mustard croutons and drizzle with more oil. Do ahead: Soup can be made 3 days ahead. Let cool; cover and chill. Reheat soup over medium-low, adding a splash of water to thin if needed.