Prepare the Pasta & Veggies: Boil a large pot of water and cook the linguine according to package directions. Two minutes before the pasta is finished, add the halved asparagus to the same pot. Drain well.
Wilt the Spinach: Return the hot pasta and asparagus to the pan. Stir in the fresh spinach, cover with a lid, and let the residual heat wilt the greens while you prepare the chicken.
Dredge the Chicken: In a large Ziploc bag, combine 3/4 cup flour, 1 1/2 tsp salt, and 1/2 tsp pepper. Add the chicken breasts and shake until evenly coated.
Sear the Chicken: Melt the butter and olive oil in a large skillet over medium-high heat. Cook the chicken until golden brown and cooked through. (Note: If your skillet is small, cook in batches, adding a bit more butter/oil for the second round).
Make the Sauce: Remove chicken from the skillet. Whisk 2 TBSP of flour into the remaining pan drippings. Gradually whisk in the chicken broth, lemon juice, and capers. Season with the remaining salt and pepper.
Combine: Toss the pasta and vegetables with the sauce.
Serve: Plate the pasta and chicken, then sprinkle with diced plum tomatoes and plenty of Parmesan cheese.