For the sauce: Heat a small saucepan over medium heat. Add the olive oil and whisk in the flour until smooth. Cook for 1 minute. Whisk in the wine, chicken broth and salt. Bring to a simmer, whisking constantly, until smooth. Reduce the heat to maintain a gentle simmer and simmer for 5 minutes. Stir in the lemon zest, juice, capers and parsley. Return to a simmer and whisk in the butter until smooth. Keep warm.
For the chicken: Preheat a grill pan over medium-high heat.
Thread one chicken strip on each bamboo skewer. Sprinkle the chicken all over with the salt and drizzle with the olive oil. Place on the grill pan and grill until golden brown and cooked through, about 2 minutes per side. Remove to a platter. Grill the lemon, cut-side down, alongside the chicken until charred, about 3 minutes. Squeeze over the cooked chicken. Serve with the piccata sauce on the side for drizzling.