Heat 1/2 tablespoon oil in a large saucepan over medium-high. Cook shallot, garlic, ginger, turmeric, and red pepper, stirring, until fragrant, about 2 minutes. Add broth, 1 cup water, and lentils. Bring to a boil over medium-high. Reduce to medium; cover and simmer, stirring occasionally, until lentils are tender, about 15 minutes. Stir in squash, and cook 5 minutes. Remove from heat, and cool 10 minutes.