Heat oven to 200C/180C fan/gas 6. Roll out the pastry to just larger than a 30cm diameter ovenproof frying pan and use the pan as a template to cut out a round of pastry. Leave in the fridge, or a cool place while you prepare everything else.
Bring a pan of water to the boil, add the carrots and parsnips and cook for 5 mins. Drain and set aside to dry off slightly. Heat the oil and 1 tbsp of the butter in your pan and cook the sliced shallots and grated parsnips for 1-2 mins over a medium-high heat until just starting to brown, then add a good splash of water to the pan, stir to scrape up any brown bits, turn the heat down slightly, cover and cook for 4-5 mins, stirring every so often. The parsnips should be tender. Add the herbs and garlic and cook for another minute, then tip into a bowl, scraping out as well as you can. Season a little and set aside.
Add the sugar and vinegar to the pan and bring to the boil. Cook for a few minutes until syrupy, then stir in the remaining butter. Set aside to cool slightly, then arrange the vegetables in the pan, fanning them out. Top with the herby shallot and parsnip mixture, pushing it into any spaces there may be. Lay the pastry over the top of the vegetables and push in the edges a little. Place in the oven and bake for 30-35 mins until puffed and deep golden brown.
While the tart bakes, put the nuts, sugar and butter in a small frying pan with a pinch of salt. Cook over a medium-high heat for about 5 mins, stirring, until the sugar has dissolved and coated the nuts. Add the rosemary for the last minute. Tip out onto a piece of baking parchment and leave to cool for a few minutes.
Once the tart is baked, remove from the oven and carefully turn out onto a serving board. Scrape out any bits that are left in the tin and add back to the tart. Bash the nuts to break them up and scatter over, along with the crumbled cheese.