Preparation Preheat oven to 400°F. Toss potatoes, 1 Tbsp. oil, 1/2 tsp. salt, and 1/2 tsp. pepper on a rimmed baking sheet. Arrange potatoes cut side down in an even layer and roast until beginning to soften and golden brown on the cut sides, 15–20 minutes. While potatoes roast, cook lentils in a medium pot of simmering salted water until lentils are tender but still retain their shape, 15–20 minutes. Drain and transfer to a medium bowl. Toss mushrooms, 1 Tbsp. oil, and 1/2 tsp. pepper in another medium bowl. Arrange around potatoes on hot baking sheet and continue to roast until mushrooms are browned and crisped and potatoes are well browned on cut sides, 12–15 minutes. Meanwhile, pulse feta, tahini, garlic, 2 Tbsp. lemon juice, half of the dill, 3 Tbsp. oil, and 1/4 cup water in a food processor until a smooth sauce forms. Add more water, 1 Tbsp. at a time, and pulse to combine until sauce is just thin enough to pour. Add scallions and remaining dill, 2 Tbsp. oil, 2 Tbsp. lemon juice, 1/2 tsp. salt, and 1/4 tsp. pepper to bowl with lentils and stir to combine. Stir in roasted mushrooms. Divide feta sauce among plates. Top with potato halves. Spoon lentil salad over and drizzle with oil.