Put the potatoes in a large pot of cold, salted water, and bring to a boil over high heat. Reduce the heat to a gentle simmer, and cook until the potatoes are tender, 20 to 25 minutes. Drain and return them to the pot.
Smash the potatoes with a fork to crush them and create lots of crevices. Add 2 tablespoons lemon juice and a healthy glug of olive oil, season with salt and pepper, and toss gently. Let them cool.
Heat the oven to 375 degrees.
Toss the cauliflower on a rimmed baking sheet with a 1/4 cup of olive oil and 1/4 teaspoon of the salt. Roast until tender and browned around the edges, 20 to 30 minutes. Add the cauliflower to the smashed potatoes.
Add the olives, onion, pickled peppers, thyme and crushed red pepper flakes, and toss gently to incorporate.
Put the feta into a food processor or blender and pulse until creamy. With the motor running, drizzle in 3 tablespoons of olive oil until well incorporated.
Add the feta to the salad and fold gently. Add the arugula and fold again. Taste, and add more salt, pepper and lemon juice, as needed.
Serve slightly warm or at room temperature, with a final squeeze of lemon over everything.