Step 1

Bring a large pot of water to a boil.

Step 2

Place walnuts in a small bowl and microwave on High until fragrant and lightly toasted, 2 to 2 1/2 minutes. (Alternatively, toast the walnuts in a small dry skillet over medium-low heat, stirring constantly, until fragrant, 2 to 3 minutes.) Transfer to a plate and let cool. Set 1/4 cup aside for topping.

Step 3

Combine the remaining 1/2 cup walnuts, parsley, garlic, salt, and pepper in a food processor. Process until the nuts are ground. With the motor running, gradually add oil through the feed tube. Add Parmesan and pulse until mixed in. Scrape the pesto into a large bowl. Add chicken.

Step 4

Meanwhile, cook pasta in the boiling water for 4 minutes. Add green beans and cauliflower; cover and cook until the pasta is al dente (almost tender) and the vegetables are tender, 5 to 7 minutes more. Before draining, scoop out 3/4 cup of the cooking water and stir it into the pesto-chicken mixture to warm it slightly. Drain the pasta and vegetables and add to the pesto-chicken mixture. Toss to coat well. Divide among 4 pasta bowls and top each serving with 1 Tbsp. of the reserved walnuts.