Place lentils into a strainer or colander; sort through and remove any shriveled lentils, debris, or rocks, then thoroughly rinse lentils under running water.
Transfer the rinsed lentils to a saucepan and pour in 2 cups water. Bring to a rapid simmer over medium-high heat, then reduce heat to maintain a very gentle simmer. (You should only see a few small bubbles and some slight movement in the lentils.)
Cook uncovered for 20-30 minutes until tender, adding more water as needed so the lentils are always just barely covered. (Older lentils may take longer to cook and may also shed their outer skins as they cook.)
While lentils cook, chop onion and mince garlic.
Heat oil in a large pan over medium.
Add 1/4 cup of onion to the hot oil; cook and stir for 30 seconds. Add half of the minced garlic to the pan, cooking and stirring an additional 30 seconds until fragrant. Set remaining onion and garlic aside.
Add 1 cup mango, vinegar, honey, 1/4 teaspoon garam masala, 1/4 teaspoon sea salt, and 2 teaspoons water to the pan. Increase heat to medium-high and bring to a boil, mashing the mango as it softens. Reduce heat and simmer for 2-3 minutes, then remove from heat and set aside.
Grate carrot. Add to a food processor along with the reserved onion and garlic, cooked and drained lentils, egg, panko, cilantro, tomato paste, remaining 1/4 teaspoon garam masala, and remaining 1/4 teaspoon salt. Pulse to combine ingredients but do not puree. Shape into 4 patties.
Add remaining 3/4 teaspoon grapeseed oil to a large skillet and heat over medium.
Cook patties 3-4 minutes per side until golden brown. Meanwhile, slice tomato.
Serve lentil burgers on rolls, topped with mango mixture, lettuce, and a slice of tomato. Serve remaining mango on the side. (Serving size: 1 patty, 1 bun)