What is the ideal approach to determine relationship between candidate predictors and a dependent variable in a data driven way? I have asked several related questions (1, 2, 3), but now I would like to ask the most basic questions and hope to get a very solid answer.
I have 40 treatment variables, and I am interested to find out which ones are related to my dependent variable. I want to do this in an entirely data-driven way. I also have two variables that I would like to control for. One of these control variables is significantly correlated with several of my predictors.
My approach at the moment is to run an adaptive LASSO, forcing in the two control variables (by setting lambda to 0 at both steps of the adaptive LASSO).

*

*Does using Adaptive LASSO make sense? If not, what approach would be better?

*Does my way of dealing with the control variables make sense? If not, how should I do it?

 A: To find out which variables have the most correlation with a dependent variable, the simplest way I can think of is to perform a Parial Least Squares (PLS) analysis in that data with your dependent variable defined as output. Then, you can obtain the VIP (Variable Importance in Projection ref, ref) values, oe even simpler to analyze that, the VIP plot, and immediately you get a visually representation of the variables with the biggest influence for the prediction of your dependent variable. USually, the variables with a VIP>1 (this treshold varies between 0.8 and 1.2) are the ones with the most impact.
You can also yield a plot representative of how each variable correlates negatively or positively with your dependent variable (of course you can also assess this through the loadings plot), so a PLS could really be a very easy way to solve your problem.
A: 
I want to do this in an entirely data-driven way

If you are interested solely in prediction then there are many different approaches including partial least squares and regularization.
However,

I have 40 treatment variables, and I am interested to find out which ones are related to my dependent variable

implies that you are thinking causally. If so, you cannot do this "in an entirely data-driven way" and hope to get sensible results. There is no way to know which variables are potential confounders or competing exposures, and should be included, and which are mediators and should be excluded.
See this answer for the kind of things that can go wrong with a data-driven procedure:
How do DAGs help to reduce bias in causal inference?
