DictionaryFormatProvider.cs

class DictFormatProvider : IFormatProvider, ICustomFormatter
{
	IDictionary<string, object> _values;
	public DictFormatProvider(IDictionary<string, object> values) {
		_values = values;
	}
	
	public object GetFormat(Type type) {
		return type == typeof(ICustomFormatter) ? this : null;
	}

	public string Format(string format, object arg, IFormatProvider formatProvider)
	{
		var value = _values[arg.ToString()];
		if (value is IFormattable formattableValue)
		{
			return formattableValue.ToString(format, null);
		}
		else {
			return value.ToString();
		}
	}

}

Program.cs

void Main()
{
	FormattableString s = $"aaa{"foo":N}bbb";
	var values = new Dictionary<string, object> {
		["foo"] = 12345.67890
	};
	s.ToString(new DictFormatProvider(values)).Dump();
}