1. Heat the oven to 400°F. Line a half-sheet-sized baking sheet with parchment paper.
2. Make the spice mixture. In a small bowl, combine the garlic powder, onion powder, and paprika.
3. In a medium bowl, combine the halved baby potatoes, half of the spice mixture, 1 tablespoon of the olive oil, salt, and pepper. Toss to coat and then transfer to the baking sheet. Spread the potatoes out and then slide the baking sheet into the oven. Set a timer for 20 minutes.
4. Place the green beans in the medium bowl and toss them with the remaining spice mixture, remaining tablespoon of olive oil, salt, and pepper.
5. When the 20-minute timer goes off, pull out the baking sheet and give the potatoes a little flip and toss. Add the green beans to the baking sheet and spread everything out in a single layer. Slide the baking sheet back into the oven and set a timer for 20 minutes.
6. Rinse out the medium bowl and give it a wipe. To the bowl, add the vegan pesto, lemon juice, and Tamari. Whisk to combine.
7. Once the potatoes are golden brown and green beans are tender, remove from the oven. Pour the pesto mixture over top and quickly toss to coat. Serve green beans and potatoes warm with chopped fresh basil and toasted pine nuts.