Preheat oven to 350 degrees F. In a small bowl toss together walnut halves, 2 teaspoons olive oil, and 1/2 teaspoon salt. Place walnut mixture in a shallow baking pan. Bake 7 to 9 minutes or until walnuts are toasted; set aside.
Meanwhile, bring a large pot of salted water to boiling. Add the beets. Return to boiling. Add pasta and cook according to package directions. Reserve 1/2 cup pasta water. Drain pasta and beets.
In an extra-large skillet heat 2 tablespoons olive oil over low heat. Add onion and garlic; cook and stir until fragrant. Stir in lemon juice, honey, and crushed red pepper. Increase heat to medium. Add pasta mixture; toss until coated (add pasta water, a little at a time, if pasta seems dry). Season to taste with salt. Remove from heat. Toss with walnuts and dill weed. If desired, serve with additional honey and crushed red pepper.