Remove the thick inner stem from the cauliflower, slice into thick 1-inch slabs.
Preheat a cast-iron grill pan over high heat.
Rub the cauliflower with olive oil and arrange the pieces in a single layer in the preheated pan.
Grill the cauliflower until charred on the outside and crisp-tender inside, about 4 to 5 minutes, flipping halfway through.
Transfer the cauliflower to a cutting board, chop into bite-size pieces, and place in a large mixing bowl.
Peel and dice the potato into 1⁄2-inch pieces, transfer to a large saucepan, cover with 4 inches of cold water, and bring to a rolling boil over high heat. Reduce the heat to medium and cook the potatoes for 6 to 8 minutes, or until almost tender.
Stir in the peas and cook another minute. Reserve 1⁄4 cup of cooking water, then drain the potatoes and peas and rinse with cold water. Add these vegetables to the cauliflower.
In a food processor, pulse together the pesto ingredients plus 2 tablespoons of the potato cooking water until smooth. Spoon onto the vegetables, add the nuts, and combine thoroughly.
Cover and chill for 10 minutes to blend flavors. Serve the salad slightly chilled or at room temperature. Garnish each serving with a few nuts.