Place the potatoes in one saucepan and the rutabaga in another, cover both with cold water, season with salt, bring to a boil, then simmer until fork tender, cooking the rutabaga 5–10 minutes longer.
Drain the vegetables, combine them in one pot, and mash with butter, initial salt, pepper, and 2 tablespoons of heavy cream until smooth, adding more cream as needed; taste and adjust seasoning.
Heat olive oil in a large skillet over medium heat, cook the onion with a pinch of salt until softened, then add the celery and carrots and cook until just tender.
Add the mushrooms and cook until they release their juices and begin to brown, then stir in the garlic and thyme and cook until fragrant.
Stir in the tomato paste, Worcestershire sauce, remaining salt, and pepper, coating the vegetables, then sprinkle over the flour and cook for about 1 minute.
Slowly pour in the vegetable broth while stirring, simmering until the mixture thickens into a gravy, then fold in the lentils and parsley and cook a few more minutes.
Preheat the oven to 425°F (220°C). Transfer the filling to a casserole dish or leave it in an oven-safe braiser, smoothing it into an even layer.
Spoon the root vegetable mash over the filling, gently spreading it to cover, leaving some peaks for browning.
Bake the Gardener’s Pie for 15–20 minutes, until the top is lightly golden and the filling is bubbling at the edges, then let it rest 5–10 minutes before serving.