Preheat your oven to 425°F. Toss the halved potatoes with olive oil, salt, and black pepper until evenly coated.
Arrange them cut-side down on a baking sheet. Roast for about 15 minutes to begin developing a golden crust underneath.
Scatter the sliced red spring onions over the partially roasted potatoes. Return the pan to the oven.
Continue roasting for about 25 more minutes until the potatoes are tender and caramelized. The onions should be lightly charred and sweet.
Add arugula, basil, chives, garlic, pine nuts, pecorino, lemon zest, lemon juice, and honey into a food processor. Pulse until roughly chopped.
With the processor running, slowly drizzle in olive oil until the pesto becomes smooth and creamy. Taste and adjust seasoning if needed.
Let the roasted potatoes cool for about 10 minutes so they’re warm but not steaming hot. Transfer them to a large bowl.
Add fresh arugula and shaved radishes. Spoon the pesto over the top and gently toss until everything is evenly coated. Serve warm or at room temperature.