Preheat oven to 220°C/425°F. Add the turnips, potatoes, parsnips and carrots to a large mixing bowl. Drizzle over 1 tablespoon of the olive oil and season generously with salt and pepper. Mix to combine and add to a parchment-lined baking sheet.
Move to the oven and roast for 10-15 minutes - until beginning to brown and the vegetables are lightly tender but still offer resistance when pricked.
Meanwhile, add the remaining oil to a large saucepan and set over medium heat. Once shimmering, add the shallot and celery along with a generous pinch of salt. Cook, stirring occasionally, until softened - about 5 minutes.
Add the garlic and miso paste and cook, stirring frequently, until very fragrant - only about 2 minutes.
Add the parsley, thyme and flour. Cook, stirring constantly, for about 3 minutes to toast the flour.
Pour over the wine and cook until the strong alcoholic smell has dissipated and the wine has fully absorbed into the flour.
Stirring constantly to avoid lumps, stream in the vegetable stock and milk. Add the roasted vegetables, kale and lemon zest. Stir to combine and bring to a simmer. Cook for about 5 minutes, just to soften and wilt the kale. Off heat, stir in the lemon juice and taste to adjust for seasoning.
On a well-floured work surface, roll out the puff pastry until it's about 5mm thick and wide enough to cover the surface of a 20cm (8in) square baking dish - or other similarly-sized baking dish. Cut it to size if necessary.
Pour the vegetable filling into the baking dish and then top with the puff pastry. Brush the pastry with the beaten egg and season the pastry with black pepper and flaky salt, if desired.
Set the pie on a rimmed baking sheet (to catch drips) and move to the oven. Bake until the filling is bubbling and the pastry is deeply golden brown, flaky and crisp - about 15-20 minutes. Allow to cool for 5-10 minutes before serving.