Heat 30ml olive oil in a saucepan over medium-high heat, add onion and garlic, stir occasionally until tender (5-6 minutes), add cavolo nero, cook until just wilted (3-4 minutes). Add tomato and 60ml water, season to taste, cook until thick (8-10 minutes). Remove from heat, stir in vinegar, cool.
Meanwhile, combine remaining oil and rosemary in a small saucepan, cook over low heat until infused (4-5 minutes), set aside.
For piadina dough, combine flour, bicarb soda and 1 tsp salt in the bowl of an electric mixer, add lard and with your fingertips, rub to combine. Add oil and 250ml water, mix on lowest speed with a dough hook until combined (2-3 minutes). Dough should come together easily and not be too dry; if it is, add extra water. Increase speed to high, knead until smooth and elastic (4-5 minutes). Divide dough into eight, form into balls, roll each on a lightly floured surface to 3mm thick. Cover with a damp tea towel to prevent drying out.
Preheat 2 tsp rosemary oil in a frying pan over high heat. Add one dough disc and cook until light golden in parts (1-2 minutes), turn and spread with a little cavolo nero mixture and scatter with an eighth of the cheese. Cook until underside is light golden in parts and cheese starts to melt (1-2 minutes). Season to taste, fold in half, turn and cook for another minute and transfer to a plate. Repeat with remaining ingredients, drizzle with a little extra rosemary oil and serve immediately.