In a large skillet of simmering water, cook the asparagus until crisp-tender, 3 minutes; drain. On a large plate, toss the asparagus with the tarragon, lemon zest, 2 tsp. EVOO and pepper. Divide into 4 bundles, then wrap each bundle with minutes 2 slices of prosciutto. In a large nonstick skillet, heat the remaining 1 tsp. EVOO over medium-high. Add the asparagus bundles and cook until the prosciutto is browned and crisp, about 3 minutes; transfer to a plate.
Season the cutlets with salt and pepper. In the same nonstick skillet , heat 2 tbsp. EVOO, two turns of the pan, over medium-high. Add half the cutlets and cook until light golden and cooked through, about 3 minutes per side; transfer to a platter. Repeat with the remaining 2 tbsp. EVOO and cutlets. Add the lemon slices to the skillet and cook until browned, about 2 minutes per side. Transfer to a plate. Add the garlic and capers; stir 1 minute. Add the wine and cook until the liquid is reduced by half, about 1 minute. Add the stock and the flour-coated butter; swirl the butter in the skillet until melted. Simmer until the sauce thickens slightly, 1 to 2 minutes. Add the parsley and lemon juice to taste.
Divide the chicken and the asparagus bundles among plates. Top the chicken with the sauce and the lemon slices.