Drill holes into the bottom of a round plastic storage container (approximately 6 inches wide and at least 4 inches high) and set aside. Line a colander with cheesecloth and set aside.
Put the milk and half-and-half into a large pot over medium heat and cook, stirring frequently, until it reaches 195 degrees F. (There will be a slight simmer and the top will be very foamy.) Remove from the heat and slowly stir in the vinegar. Put the colander into the sink and pour this mixture into the cheesecloth. When most of the liquid has passed through, add the tomatoes, basil, and salt and stir gently to incorporate. Gather up the ends of the cheesecloth and transfer the cheese to the plastic container. Set the container on a rack on a sheet pan to catch the whey. Put a plate on top and weight it down with some heavy cans to squeeze out the excess liquid. Let rest for 1 hour, remove the cheesecloth, and return it to the plastic container with the plates and weights. Put it into the refrigerator overnight. To serve, put the cheese onto a plate and brush with a little olive oil.