High-mindedly is the adverb of high-minded. Someone who speaks high-mindedly talks from those high moral principles — about what people *ought* to do. It can sound admirable, or it can sound as if they have forgotten how ordinary life actually works. Same split feeling as high-minded: praise in one mouth, a raised eyebrow in another. It is a little formal. You use it with a verb: argued high-mindedly, wrote high-mindedly, refused high-mindedly.