Combine 4 tablespoons oil, garlic, anchovies, red pepper flakes, and 1/2 teaspoon salt in small bowl; set aside
Heat remaining 2 tablespoons of oil in 12-inch nonstick skillet over medium-high heat until shimmering. Add cauliflower and ¼ teaspoon salt; cook, stirring occasionally, until well browned and tender, 10 to 12 minutes.
Push cauliflower to sides of skillet to create 3-inch clearing; add oil-garlic mixture and cook, stirring constantly, until fragrant, about 1 minute. Stir to combine garlic mixture with cauliflower, add the dried figs and cook for 1 minute longer. Remove skillet from heat and stir in walnuts, parsley, and lemon juice.
Meanwhile, bring 4 quarts water to boil in large Dutch oven. Add pasta and 1 tablespoon salt to boiling water; cook until al dente. Reserve ¾ cup pasta cooking water, drain pasta, and return pasta to Dutch oven.
Add sauce and reserved cooking water to pasta and toss to coat. Season with salt to taste. Transfer to serving bowl, sprinkle with cheese, and serve immediately.