Ten years old is old enough to learn how to cook, and as such I'd suggest that as the starting point for your approach to this.
I'm not talking 'tuna surprise and PB&J', but help her learn the fundamentals of cooking, perhaps from a cooking school if you two don't have enough of a background to teach that. Learn the fundamentals of preparation, how different ingredients are prepared, and how to prepare a complete meal.
Learning to cook at this level serves several purposes.
First, it allows her to be involved in the meal preparation process. This both means she can prepare meals, which mean she is herself responsible for any taste issues - but also she may begin to understand how you feel when she acts like she is unhappy with your food. It's also a great thing to learn for when she becomes an adult. You can sit down with her and plan menus together - hopefully letting her do most of the planning, just occasionally reminding her about details like "don't forget a vegetable side here" and "Yes, King Crab stuffed with Oysters sounds wonderful, but I don't have $100 for a dinner right now", once she's gotten used to doing it. Nothing teaches like doing.
Second, it allows her an "out" that is a fair one. You prepare dinner, and let her know in advance what it is. If she doesn't like it? She can prepare something of her own for herself - as long as you approve it from a nutritional point of view, and work out any cost issues (ie, if it's costing you more for food to do it this way, that comes out of her allowance in some way and/or she picks cheaper food items). If she prepares enough for several meals, that can become leftovers for the next time she has a problem with dinner, or for lunch, or whatever.
Third, it may help her develop a better understanding of what she likes - and hopefully a much broader palate. I wasn't a picky eater as a child, but I did have particular taste preferences that I didn't really understand. I thought I hated onions, for example, to learn that I only disliked raw onions; cooked onions are wonderful. Learning technique involves a lot of learning how individual ingredients are cooked, and how they taste both uncooked and cooked. When I made the effort to learn the art of cooking (so far self taught, but hopefully at some point in a more formal way), I learned a lot about my tastes, and am much more able to explain what I like and don't like - and learned to like many more things than I used to.
Finally, if this is primarily a psychological issue, this should help significantly. Not only will she have the tools to understand herself (if it's an unconscious one), but she won't have the excuse anymore. Don't like what we cooked? Fine, go make something yourself. That's pretty hard to argue with, when you are capable of making something yourself perfectly well, and have sufficient ingredients and time for it. I suspect you'll find that she starts complaining less about the food you make - or cooks more, one of the two - and everyone will be happier.
And as far as restaurants, I would let her simply eat or not eat at her own discretion there. Unless you're eating out all the time, one meal every so often of poor quality or quantity won't be the end of the world. Enforce "What" rules - ie, no, you can't order ice cream for your entree - and let her work out how much, and if that amount is nothing, fine - don't worry about it. She can make something when she gets home.
As an aside - while it's possible your child is "just a picky eater" whose problem is "psychological" in nature, I would resist making such assumptions entirely. Some people are "super tasters", capable of differentiating far more than most in terms of the taste of their food. This can make them seem very picky, because they'll dislike certain tastes that you don't really notice - especially if they dislike foods that are fairly common. Sulphur, for example, is a common "ick" among super tasters, and is present in a lot of foods - onions, broccoli, cabbages, etc. It's hard to say without knowing her and knowing more about her what exactly is going on here, but it's a bad idea to dismiss her feelings entirely (on any issue, but this one for sure) assuming it's a behavior problem and not something legitimate that she perhaps can't explain clearly.