Heat oil in large skillet until shimmering then add onion and carrot. Saute until beginning to soften, 2-3 minutes.
If using mushrooms add now with salt and pepper and saute until they release liquid. Add remaining vegetables (do not add meat now) and saute about 5 minutes.
Sprinkle vegetables with flour and continue cooking until flour browns and is incorporated with vegetables, about 2 minutes. Add liquid and thyme, bring to boil and allow to simmer until thickened and vegetables are just tender.
Taste and adjust for seasoning.
Add meat (if using), peas, scallions and herbs.
Place in baking dish - 9" pie plate, 8" square baking dish, cast iron skillet, individual ramekins if you're feeling fancy. Cover with topping of choice, poke a few holes in top to allow steam to vent and bake, on a baking sheet in case of overflow, according to topping directions. If using a biscuit topping we find it works best to cook the biscuits for about about 10 minutes on a baking sheet (have the potpie in the oven at the same time) and then place them on the pie filling.
If making ahead, refrigerate filling until cool then cover with foil or plastic wrap. Add topping just prior to baking and increase cooking time by 10-15 minutes to account for the cool start.