Put the potatoes in a large pot of cold water and add salt until it tastes like the sea. Bring to a boil, reduce the heat to a gentle simmer and cook until tender, 20-25 minutes. Once the potatoes are cooked, drain them and return to the pot.
Smash the potatoes with a fork to crush them and create lots of crevices, add 2 tablespoons of the lemon juice and a healthy glug of olive oil, season with salt and black pepper and toss gently. Let them cool.
Heat the oven to 375.
Put the cauliflower in a bowl and add 1/4 cup olive oil and 1/2 tsp salt – toss to coat all the florets and spread them over a rimmed baking sheet. Roast until tender and nicely browned around the edges, 20-30 minutes. Add the cauliflower to the smashed potatoes.
Add the olives, onion, pickled peppers, thyme, and chili flakes tot he potatoes and cauliflower. Toss gently, season with more salt and black pepper and some olive oil and a bit more lemon and toss again.
Put the feta in a food processor and pulse until creamy. With the motor running add 3 tablespoons of olive oil. Add the feta tot eh salad and fold gently. Add chopped arugula and fold again. Taste and season accordingly. Serve slightly warm or at room temperature with a final squeeze of lemon over everything.
