Preheat oven to 400°F.
Cut puff pastry into small squares and place on a parchment-lined baking sheet.
Whisk egg with water and brush over pastry. Sprinkle with salt and thyme.
Bake 12–15 minutes until puffed and golden.
Meanwhile, melt butter with olive oil in a large soup pot.
Add onion, carrots, and celery and cook 5–7 minutes.
Add garlic and cook 30 seconds.
Stir in flour and cook 1–2 minutes.
Slowly whisk in chicken broth.
Add potatoes, thyme, sage, salt, and pepper.
Simmer 12–15 minutes until vegetables are tender.
Add chicken, peas, and corn and cook 3–4 minutes.
Reduce heat to low and stir in half-and-half, cream, Parmesan, and Dijon.
Warm gently for 2–3 minutes.
Stir in parsley and adjust seasoning.
Serve topped with puff pastry croutons.