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 rimmed baking sheet, toss broccoli, garlic 1 Tbls olive oil; season with salt and pepper. spread in a single layer; roast, turning sheet halfway through, until tender and beginning to brown, 15 to 20 minutes. Set aside.
Meanwhile cook pasta in a large pot of boiling water, according to package instrutions. Drain pasta, reserving 1/4 cup cooking water; rinse under cold water until cool. Drain and return to pot.
Remove skin from roasted garlic. In a small bowl, mash garlic until forms a paste; whisk in lemon juice and remaining 1 tbls of olive 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.