Heat olive oil in a large saucepan. Add carrots, celery, onion, and garlic and cook until vegetables soften.
Stir in lentils and coat with oil. Add chicken stock to cover by about 1 inch. Bring to a boil, reduce heat, and simmer for 25-35 minutes, until lentils are soft.
Season with salt and pepper. Set aside but keep warm.
For the beurre blanc:
In a small saucepan, bring shallots, garlic, fennel seed, bay leaf, peppercorn, white wine, and rice vinegar to a boil and cook until reduced by half, about 15 minutes. Strain through a fine mesh sieve and discard solids.
Pour liquid back into saucepan and put onto very low heat. Whisk in butter, a couple cubes at a time, until butter is melted and sauce is emulsified. Whisk in rosemary and lemon juice. Season with salt and pepper to taste and set aside.
For the trout and assembly:
Heat olive oil in a large skillet over medium high heat. Season fish with salt and pepper and place skin side down onto pan. Cook until skin gets nice and crispy, about 4 minutes. Flip trout and cook for another 2-4 minutes, depending on the thickness.
Place trout on top of lentil mixture and top with beurre blanc.
Recipe by Things I Made Today at https://www.thingsimadetoday.com/2014/04/28/roasted-trout-beurre-blanc-lentils/