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