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Pour water into a large saucepan to cover lentils by 4". Bring to a boil, reduce heat to medium, and simmer, stirring occasionally, until lentils are tender but not mushy, about 10 minutes from the time the water reaches a boil. Drain very well in a fine-mesh sieve, then let cool at least 10 minutes (spread them out on a baking sheet, which will help them cool faster).
Meanwhile, combine yogurt, lemon zest, lemon juice, and half of garlic in a small bowl. Season with salt; set aside.
Finely chop mushrooms and transfer to a medium bowl. Add miso, paprika, 2 Tbsp. oil, and remaining garlic clove. Add lentils (you should have 2 cups cooked) and stir and mash them with the back of a spoon or a potato masher until lentils are partly mashed but with lots of whole lentils still remaining. Vigorously stir in flour until mixture holds together when squeezed; if it doesn’t, continue to mash until it does and add 1–2 Tbsp. flour if needed. Form into 6 patties about ¾" thick.
Working in 2 batches, heat 1 Tbsp. oil in a large nonstick skillet (nonstick is essential since these will definitely stick to a regular skillet) over medium. Cook patties until deeply browned and very crisp on bottom side, about 3 minutes. Carefully turn and repeat on second side, adding more oil as needed to maintain a light coating around patties in skillet.
Spread reserved yogurt mixture on buns. Top with burgers, sprouts, and pickles.
Do Ahead: Patties can be made 4 days ahead. Tightly wrap in plastic and chill.