Heat oven to 350°F. Slice garlic head in half crosswise and set cut side up on a piece of aluminum foil. Drizzle with 2 Tbsp oil and 2 Tbsp cold water; wrap into a tight package. Bake until tender, 1–1 ½ hours.

Heat remaining oil and the sliced garlic in a 1-qt. saucepan over medium; cook until garlic is golden, 4–6 minutes. Using a slotted spoon, transfer garlic chips to paper towels to drain; set ⅓ cup oil aside. Reserve remaining oil for another use, if you like.

Heat reserved ⅓ cup oil, the minced garlic, and chile flakes in a 14” high-sided skillet over medium. Cook until garlic is soft but not golden, 2–3 minutes. Add pasta and stock; bring to a boil. Cook, using tongs to stir pasta occasionally, until liquid is almost evaporated and pasta is al dente, about 12 minutes. Unwrap roasted garlic and squeeze cloves into pasta. Add parmesan, lemon juice, parsley, and salt; toss to combine. Garnish with reserved garlic chips.