Preheat the broiler. Toss the broccolini with 1 tablespoon olive oil, 1/4 teaspoon salt and a few grinds of pepper on a baking sheet; add the lemon slices. Broil until the broccolini is tender and just starting to char, 5 to 6 minutes. Set aside and cover to keep warm.
Meanwhile, heat 1 tablespoon butter and the garlic in a nonstick skillet over medium-high heat. Add the mushrooms and cook until browned and the liquid is evaporated, about 6 minutes. Transfer to a bowl.
Pound the turkey to an even thickness using a heavy skillet, if needed. Mix the flour, 1/2 teaspoon salt and a few grinds of pepper in a shallow dish. Dredge the turkey in the flour. Heat 1 tablespoon olive oil in the skillet over medium-high heat. Add half of the turkey and cook until browned and cooked through, 2 to 3 minutes per side; transfer to a plate. Repeat with the remaining 1 tablespoon olive oil and turkey; transfer to the plate.
Increase the heat to high. Add the wine and cook until reduced by half, 2 minutes. Add the broth and cook until reduced by half, 4 minutes. Add the mushrooms and the remaining 1 tablespoon butter. Remove from the heat and stir in the parsley and 1 tablespoon lemon juice. Spoon the sauce over the turkey; serve with the broccolini.