To make the ragù, heat olive oil in a medium skillet over medium heat. Add the carrot, onions and garlic, and sauté until soft and just starting to brown, about 6-7 minutes. Stir in the tomato paste and continue to cook for another 30 seconds. Add stock, lentils and fennel pollen, and season with salt and pepper, to taste. Bring to a boil, cover and cook until lentils are soft and cooked through, about 35-45 minutes.
While the lentils are cooking, poach the eggs. Fill a medium saucepan two-thirds full of water and bring to a simmer. Add vinegar and salt to the water. Crack one egg into a small bowl and gently tip it into the simmering water. Cook until the whites are set and the yolk is still runny, about 4 minutes. Remove with a slotted spoon and place onto a paper towel-lined plate. Repeat with the remaining eggs.
To serve, nestle the poached eggs in lentil ragù and garnish with the hot peppers in olive oil and the chopped mint. Serve hot.