Use a rasp-style grater to grate the cheeses.
Heat oil in Dutch oven over medium-high heat until just smoking. Add broccoli, ½ teaspoon salt, and pepper and cook, stirring occasionally, until broccoli is crisp-tender and dark brown in spots, about 8 minutes. Stir in half of garlic and cook until fragrant, about 30 seconds; transfer to bowl.
Combine water, half-and-half, pasta, remaining 1¼ teaspoons salt, and remaining half of garlic in now-empty Dutch oven and bring to boil over medium-high heat. Reduce heat to medium and simmer, stirring often, until pasta is tender, 15 to 18 minutes.
Off heat, stir in cheeses, lemon juice, and broccoli until combined. Cover and let sit until sauce is thickened, about 5 minutes. Stir and serve.
