Place the walnuts in a dry skillet over medium heat and toast for 3–4 minutes, stirring frequently, until fragrant and lightly golden. Watch them carefully — walnuts go from perfectly toasted to burnt very quickly. Remove from heat and let cool for a minute.
Bring a large pot of water to a boil. Cook the pasta according to package instructions until al dente. Reserve a cup of pasta cooking water before draining. Drain and set aside.
Add the toasted walnuts, fresh spinach, olive oil, lemon juice, garlic, parmesan, and black pepper to a blender or food processor. Blitz until smooth and vibrant green. If too thick, add a splash of reserved pasta water and blitz again. Taste and adjust lemon or pepper as needed.
While the pasta cooks, heat the teaspoon of olive oil in a small skillet over medium-high heat. Add the halved cherry tomatoes cut-side down and cook for 4–5 minutes without stirring — you want them to blister and caramelise on the bottom.
Add the drained pasta to a large bowl. Spoon the walnut pesto over the pasta and toss well, adding splashes of reserved pasta water to reach a silky, coating consistency. Divide between two bowls, top with the blistered cherry tomatoes, and finish with a drizzle of extra virgin olive oil and a few extra walnut pieces. Serve immediately.