Heat the oven to 400°F. Set out a 10-inch oven-safe skillet so it is ready for both the stovetop and the oven.

Unfold the puff pastry on a lightly floured counter or cutting board. Let it sit at room temperature while you make the filling so it softens enough to drape over the skillet later.

Pour the olive oil into the skillet and set the skillet over medium heat.

Add the diced onion, carrots, and celery to the skillet. Stir to coat them in the oil.

Cook the vegetables for 5 to 6 minutes, stirring now and then, until the onion looks soft and the carrots start to lose their raw crunch.

Add the sliced mushrooms to the skillet. Stir them into the vegetables.

Cook the mushrooms for 6 to 8 minutes, stirring every minute or so, until they release their liquid and the skillet looks mostly dry again.

Sprinkle the flour, dried thyme, kosher salt, and black pepper over the vegetables in the skillet.

Stir for 1 minute, until the flour disappears into the vegetables and no dry patches remain.

Pour the vegetable broth into the skillet a little at a time, stirring after each addition. Scrape the bottom of the skillet with a wooden spoon so any browned bits mix into the sauce.

Pour the milk into the skillet. Stir well and let the mixture come to a gentle simmer.

Simmer for 2 to 3 minutes, stirring often, until the sauce thickens enough to lightly coat the spoon.