Bring water to a boil in a large saucepan. Stir in lentils, reduce heat to medium-low and simmer until very tender and beginning to break down, about 25 minutes for brown lentils or 30 minutes for green lentils. Drain in a fine-mesh sieve.

Meanwhile, heat 1 tablespoon oil in a large skillet over medium heat. Add carrot, shallots and celery and cook, stirring, until softened, about 3 minutes. Add almonds, thyme, salt and pepper; continue cooking until the almonds are lightly browned, about 2 minutes. Transfer the mixture to a food processor; add 1 cup of the cooked lentils. Pulse several times, scraping down the sides once or twice, until the mixture is coarsely ground. Transfer to a large bowl; stir in the remaining lentils. Let cool for 10 minutes. Mix in egg yolk and lemon juice. Cover and refrigerate for 1 hour.

Form the lentil mixture into 5 patties. Heat the remaining 1 tablespoon oil in a large nonstick skillet, preferably cast-iron, over medium-high heat. Add the patties and cook for 3 to 4 minutes. Turn gently and continue to cook until lightly browned and heated through, 3 to 4 minutes more. Serve immediately.

Note: French green lentils are smaller and firmer than brown lentils. They cook more quickly, too, in about 20 minutes. They can be found in natural-foods stores and some larger supermarkets.

MAKE AHEAD TIP: Prepare through Step 2 up to 6 hours in advance.