Combine yoghurt, chutney and salt and pepper in a bowl. Cover and refrigerate. Run a vegetable peeler along the length of cucumbers to form ribbons. Place in a bowl. Cover and refrigerate.
Heat 1 tablespoon oil in a non-stick frying pan over medium heat. Add onion and carrot. Cook, stirring occasionally, for 3 minutes or until soft. Add curry paste. Cook, stirring, for 1 minute or until aromatic. Remove from heat.
Place lentils in a food processor. Process until roughly chopped. Transfer to a large bowl. Add carrot mixture, breadcrumbs, coriander and salt and pepper. Mix until well combined. Using clean hands, shape mixture into four 2cm-thick patties. Place on a plate. Cover and refrigerate for 30 minutes.
Heat butter and remaining oil in a large, non-stick frying pan over medium heat. Cook patties for 2 to 3 minutes each side or until golden and heated through. Top bread roll bases with lettuce, patties, cucumber and yoghurt sauce. Cover with bread roll tops. Serve.