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Preheat oven to 375 degrees.
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Cut 1 1/2 inches from the tip of the asparagus; set 16 pieces aside. Cut asparagus stalks in half crosswise. Cut the bottom halves into 1/2-inch pieces and set aside; you should have about 2 cups. Halve remaining half crosswise and chops; you should have about 1 cup. Set aside separately.
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Prepare an ice-water bath and set aside. Heat olive oil in a medium saucepan over medium-high heat. Add coarsely chopped asparagus bottoms and season with salt; cook for 1 minute. Add 1 cup water and continue cooking until water is almost completely evaporated, 3 to 5 minutes. Add milk, cream, and spinach; cook until spinach is wilted, about 2 minutes.
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Transfer asparagus mixture to a bowl and set in ice-water bath; let cool completely. Transfer to the jar of a blender or the bowl of a food processor; blend until smooth. Add eggs and yolk; season with salt and blend until combined.
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Set 16 1/4-cup heatproof glass bowls in a roasting pan; divide asparagus mixture evenly between bowls. Set roasting pan on oven rack and carefully add enough hot water to roasting pan to come halfway up the sides of the bowls. Cover with parchment-paper-lined aluminum foil and bake until asparagus mixture is set, about 18 minutes. Remove flans from roasting pan and transfer to refrigerator until chilled.
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Meanwhile, bring a medium pot of water to a boil over high heat. Prepare an ice-water bath and set aside. Add reserved asparagus tips to boiling water and cook until tender-crisp, about 3 minutes. Using a slotted spoon, transfer to ice-water bath to cool. Remove from water bath and pat dry; halve tips lengthwise and transfer to a small bowl; set aside. Add finely chopped asparagus to boiling water; cook until bright green in color, about 3 minutes. Drain and transfer to ice-water bath to cool. Drain and transfer to a medium bowl; set aside.
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Bring another medium pot of water to a boil; add white vinegar. Prepare an ice-water bath and set aside. Carefully crack quail eggs into a large bowl, taking care not to break the yolks. Using a spoon, stir boiling water to create a swirling motion and carefully add eggs to water. Cook until whites are set, about 1 1/2 minutes. Drain and transfer eggs to ice-water bath to cool. Trim whites using kitchen shears so that you have 16 individual poached quail eggs; season with salt.
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When flans have chilled, drizzle asparagus tips and finely chopped asparagus with olive oil and 1/2 teaspoon sherry vinegar each; season with salt and pepper and stir to combine. Divide finely chopped asparagus evenly among flans. Top each with a quail egg and two asparagus spears. Using kitchen shears, snip each quail egg to expose yolks; serve.
