1. In medium saucepan, bring water mixed with 1/2 teaspoon salt to boil. As soon as the water comes to a boil, add asparagus and boil until tender, about 3 minutes. Drain. Set aside until needed.
2. Lay chicken breasts between sheets of plastic wrap and pound to 1/2 inch thickness. On a medium plate mix flour with 1/4 teaspoon salt and pepper; dredge chicken to coat, patting off any excess.
3. In medium, non-stick frypan over medium heat, warm 1 teaspoon butter and oil. Saute chicken breasts on both sides until browned and firm, about 3 minutes per side. Remove chicken; keep warm until ready to serve. Add onion to skillet and saute until softened, about 4 minutes. Increase heat to high, add tomato and saute until most of the moisture has evaporated. Lower heat to low and add crab meat; toss gently, just until hot.
4. To serve, set one chicken breast on each plate; arrange 5 asparagus spears on top of each. Spoon some of crab mixture over asparagus and drizzle with Lemon Chive Sauce. Garnish with cherry tomatoes.
5. Lemon Chive Sauce: In small saucepan over low heat, melt 1 tablespoon butter with lemon juice to make sauce. Whisk in cream cheese, milk and chives until smooth