Preheat the oven to 375°F (190°C). Spread the walnuts in a baking pan. Toast until they turn a shade or two darker and are fragrant, 6–8 minutes. Pour the nuts onto a plate to cool.
In a food processor, combine the toasted walnuts, lemon zest, and garlic and pulse just to combine. Add the basil and arugula and process until coarsely chopped. With the motor running, slowly pour in 4 Tbsp (2 fl oz/60 ml) of the oil. Continue to process until the mixture is moist and well blended but still slightly chunky. Transfer the pesto to a small bowl and adjust the seasonings with salt and pepper.
In a frying pan, warm the remaining 1 Tbsp oil over medium-high heat. Add the tomatoes and a pinch of salt and sauté until the tomatoes are warmed through and their skins are just beginning to split, 3–4 minutes. Remove from the heat and stir in the pesto. Transfer the tomatoes to a serving dish and crumble the cheese over the top. Serve hot or at room temperature. If desired, top the tomatoes and cheese with a scattering of arugula leaves.