Dataset with 10% missing without either using a multiple imputation technique or removing the samples. How would I address the missingness? I have a dataset comprising of 10% missing fraction where missingness can be predicted from the data which I believe is Missing Not At Random. How would I address the missingness of a dataset without using a multiple imputation technique or case wise deletion or removal of samples with missing?
 A: Missing not at random (NMAR) has a very specific meaning in this literature: it means that the chance of being missing depends on the unobserved value of the variable itself. 
Missing at random (MAR) means that the chance of being missing does not depend on the unobserved value of the variable itself, but it can depend on other observed variables.
Only Missing completely at random (MCAR) means that the chance of being missing does not depend on anyhting.
You state that your data is NMAR and that you can predict the missing values from the data. That is bold statement. I suspect that you mean that the data is MAR. 

Anyhow having said that, if you don't want to do MI and don't want to remove observations, then all that is left is to use a model that includes the missing data mechanism (typically using maximum likelihood). What that model looks like depends on the exact problem. (weighting is also sometimes used, but that also removes the observations from the data, so that does not meet your criteria) You can read more in (Little & Rubin 2002)
Roderick J. A. Little and Donald B. Rubin (2002) Statistical Analysis with Missing Data, Second Edition. Wiley.
