Preheat oven to 450 degrees. Trim 1 inch from stems of broccoli. Peel outer layer of stems; cut diagonally into 1/2-inch slices. Separate florets into bite-size pieces. On one large or two small rimmed baking sheets, toss broccoli, garlic, and 1 tablespoon oil; season with salt and pepper. Spread in a single layer; roast, rotating sheets and turning broccoli halfway through, until tender and beginning to brown, 15 to 20 minutes. Set aside.
Meanwhile, cook pasta in a large pot of boiling salted water, according to package instructions. Drain pasta, reserving 1/4 cup cooking water; rinse under cold water until cool, and drain. Return to pot.
Remove skin from roasted garlic. In a small bowl, mash garlic until it forms a paste; whisk in lemon juice and remaining tablespoon oil. Season with salt and pepper. Pour mixture over pasta; toss in broccoli, red peppers, olives, and walnuts. Add some reserved cooking water, if needed. Serve at room temperature or chilled.