Heat a large skillet over medium heat and add a tablespoon of oil. Once shimmering, add finely diced onion and carrot, cooking gently until they soften and turn translucent, about 5 minutes, filling your kitchen with a sweet, savory aroma.
Add smoked paprika and ground cumin to the vegetables, stirring for about 30 seconds until fragrant, creating a warm, smoky scent that fills the air.
Stir in the cooked lentils and tomato paste, mixing until the lentils are coated and the paste darkens slightly, about 1 minute, building a rich base for your sauce.
Pour in the vegetable broth and bring the mixture to a gentle simmer. Reduce the heat to low and let it cook uncovered for 10-15 minutes, stirring occasionally until the sauce thickens and deepens in color.
Stir in apple cider vinegar, tasting and adjusting seasoning with salt or more vinegar if desired, to add brightness and balance to the smoky, savory flavors.
While the sauce simmers, slice and toast your burger buns lightly until golden and slightly crisp, adding extra flavor and texture.
Remove the sauce from heat and let it rest for a couple of minutes to allow flavors to meld and the sauce to thicken further.
Spoon a generous amount of the smoky lentil mixture onto the toasted buns, piling high for a hearty, satisfying presentation.
Serve immediately with your favorite sides or pickles, enjoying the thick, flavorful filling with a chewy, smoky aroma lingering.
