In a medium nonstick skillet, heat 1 tablespoon of olive oil over medium-high heat until shimmering. Add shallots and cook, stirring frequently, until very soft and browned, about 15 minutes; lower the heat if necessary to prevent burning. Season to taste with salt and pepper and set caramelized shallots aside. Wipe out skillet.
Press salmon filets between paper towels to dry surfaces thoroughly. Season on all sides with salt and pepper. Heat remaining 1 tablespoon in the skillet over medium-high heat until shimmering. Add salmon filets skin side-down. Immediately reduce heat to medium-low. Cook, pressing gently on back of filets to ensure good contact with skin, until skin is rendered and crisp, about 6 minutes. If skin shows resistance when attempting to lift with a spatula, allow it to continue to cook until it lifts easily. Flip salmon and cook on second side until a thermometer inserted into the thickest part registers 120°F for medium rare or 130°F for medium, about 1 minute longer. Transfer salmon to a paper towel-lined plate and allow to rest.
Add lentils, thyme or rosemary, broth, whole grain mustard, and caramelized shallots to the skillet and bring to a boil. Reduce heat to medium and simmer until the broth has reduced by about half, about 5 minutes. Remove from the heat and stir in parsley and lemon juice. Season to taste with salt and pepper. Spoon the lentils onto plates, set the salmon on top, drizzle with more olive oil, and serve immediately.
One-Pot Crispy Salmon with Mustardy Lentils