Preheat the oven to 400 degrees F.
On a cutting board, unroll the pie crusts.
Using the spiked surface of a meat tenderizer or a similar tool, lightly press a dotted design into the pie dough.
Cut the pie dough into 2-inch discs, then transfer the dough discs to a rimmed baking sheet.
Using a pastry brush, brush the tops of the dough discs evenly with 2 tablespoons of the melted butter.
Sprinkle the dough discs with the Maldon sea salt.
Bake the dough discs on the middle rack in the oven until they are golden-brown, about 6-8 minutes. Set the pie crust crackers aside.
In a Dutch oven over medium-high heat, add the remaining butter and the olive oil and cook until the oil is shimmering and the butter is melted but not browned.
Add the carrots and the celery to the oil mixture and cook, while stirring, until they start to soften, about 3 minutes.
Add the onions to the carrot mixture and cook, while stirring, until the onions are translucent, about 3-4 minutes.
Add the garlic to the vegetable mixture and cook, while stirring, until the garlic is fragrant, about 30 seconds.
Add the chicken stock to the vegetable mixture and cook, while stirring and scraping the bottom of the pan to release the browned bits, until incorporated.
Add the white wine, the peas, and the hash browns to the soup mixture and stir to combine.
Bring the soup mixture to a boil.
Reduce the heat to medium and allow the soup mixture to simmer until the flavors meld together, about 10 minutes.
Reduce the heat to low.
Add the chicken and the heavy cream to the soup mixture and stir to combine.
Season the soup with 1/2 teaspoon the salt, the black pepper, the bay leaves, the herbs de Provence, the dried basil, and the dried thyme and stir well to combine.
Allow the soup to simmer over low heat until thickened and the flavors meld together, about 30 minutes.
Remove and discard the bay leaves.
Taste the soup for seasoning and add the extra salt, as needed.
Serve the soup with the pie crust crackers.