1. First, preheat the oven according to the directions on the packaging on your puff pastry. Carefully unfold the sheet and roll out the pastry on a floured surface until it is about ⅛ inch thick. Using your serving bowl as a guide, carefully cut rounds out of the pastry about ¼ inch larger than the mouth of the bowl. Place the rounds on a parchment lined baking sheet and bake until golden brown, about 10-12 minutes. Take out of the oven and allow to cool slightly.
2. While the pastry is cooling, prepare the pot pie filling. Heat the olive oil in a large skillet over medium heat. Add the onions and carrots and cook until the onions are translucent, about 3 minutes. Add the baby broccoli, zucchini, and snap peas and cook until the broccoli is bright green and tender, about 3 minutes. Season with salt and pepper.
3. Add milk and 1¼ cups of the chicken broth, reserving ¼ cup of broth. Bring vegetable mixture to a simmer over medium high heat, about 4 minutes. Add the white chocolate and stir until melted. Turn the heat down to medium low and add the chicken and thyme and continue to simmer.
4. Make a slurry with the remaining ¼ cup of chicken broth and cornstarch. Add the slurry to the pan and stir to incorporate, mixture will begin to thicken. After about 5 minutes, turn the heat off and add the lemon juice, lemon zest, chives, and dill. Stir to combine. Check for seasoning and add salt and pepper to taste.
5. To serve, ladle generous portions of the chicken mixture into two serving bowls. Top more chives and the puff pastry lids. Serve hot.