1. Snap off and discard tough ends of asparagus; cut asparagus in half crosswise.
2. Prepare linguine according to package directions, adding asparagus during last 2 minutes of cooking. Drain; return to pan. Stir in spinach; cover and keep warm over low heat.
3. Combine 3/4 cup flour, 1 1/2 tsp. salt, and 1 tsp. pepper in a large zip-top plastic bag. Add chicken cutlets; seal bag, and shake to lightly coat.
4. Melt 1 Tbsp. butter with 1 Tbsp. olive oil in a large nonstick skillet over medium-high heat. Cook 3 cutlets in skillet 2 to 3 minutes; turn and cook 2 to 3 minutes or until lightly browned and done. Remove from skillet. Repeat procedure with remaining 1 Tbsp. butter, 1 Tbsp. oil, and 3 cutlets. (Chicken may be kept warm in a 250° oven on a wire rack.)
5. Whisk 2 Tbsp. flour into skillet, and cook 30 seconds. Whisk in chicken broth, next 3 ingredients, and remaining 1/2 tsp. salt and pepper. Cook over medium-high heat 6 to 8 minutes or until slightly thickened, whisking to loosen particles from bottom of skillet. Pour over warm pasta mixture; toss to combine. Transfer to a serving dish, and sprinkle with tomatoes. Serve immediately with chicken and Parmesan cheese.
Worth the Splurge: Use Parmigiano-Reggiano cheese. Allow your guests to grate the cheese tableside right onto their plates.