What Kind of Network should I be considering? was hoping someone can guide me in the right direction with this problem.

I have a dataset that contains the history of an experiments that lasts 40 minutes; taking a measurement every minute. I have historical data for about 600 runs. So I have a total of 24000 rows in my dataset.

I would like to set up a model in where I can train it on the different multiple runs where all 40 minutes are available; and each minute has same feature count of 11.

My plan then was to feed the model anywhere from 2-35 rows of live data, and I would like it to predict one of the values at the 40 minute mark.

Any help would be greatly appreciated.
You want to use a Recurrent Neural Network.  More specifically, look into the LSTM variant and understand it's use cases over a "regular" RNN, and go from there.  RNN's are good for capturing sequential data patterns and this is essentially what you want to do.