For chive oil and crostini:
Set a coffee filter in a glass; set aside. Blanch chives for 10 seconds in a medium saucepan of boiling salted water. Transfer to a medium bowl of ice water to cool. Squeeze chives dry; coarsely chop and place in a blender. Add 3/4 cup oil; purée until smooth. Pour through coffee filter; let drain for 3 hours at room temperature, or overnight in refrigerator (do not press on solids).
DO AHEAD: Can be made 1 week ahead. Cover; chill. Bring to room temperature before using.
Preheat oven to 400°F. Line a rimmed baking sheet with parchment paper. Brush bread slices (or toss bread cubes) with 1 tablespoon olive oil; toss with rosemary sprigs. Transfer to prepared sheet; bake until crisp, about 12 minutes. Let cool completely. Discard rosemary sprigs.
DO AHEAD: Can be made 1 day ahead. Store airtight at room temperature.
For soup:
Preheat oven to 350°F. Place whole cauliflower head in a large baking dish, rub with 4 tablespoons butter, and season with salt. Add 1/2 cup water to dish. Bake uncovered, tenting with foil if cauliflower begins to brown, until a knife inserted into the core meets no resistance, about 1 1/2 hours. Remove cauliflower from oven; let cool. Coarsely chop and set aside.
Melt 2 tablespoons butter in a large saucepan over medium-low heat. Add onion; cook, stirring occasionally, until onion is soft and translucent, about 15 minutes. Add cauliflower and 4 cups water. Simmer until cauliflower is very soft, about 10 minutes. Let cool slightly. Working in batches, purée in a blender until very smooth.
DO AHEAD: Can be made 2 days ahead. Cover; chill.
Return soup to pot and bring to a simmer, adding more water if too thick. Season with salt. Remove from heat and whisk in remaining 6 tablespoons butter and cream. Serve warm in shallow bowls. Rest crostini on edge of each bowl or scatter croutons over; drizzle with chive oil.