Bring milk, salt, and 4 cups water to a boil in a medium saucepan over medium-high heat. Gradually whisk in grits; return to a boil. Reduce heat to medium-low, and simmer, stirring occasionally, 20 to 25 minutes or until thickened. Remove from heat, and stir in cheese and butter. Cover and keep warm.
Pour water to depth of 2 inches into a large saucepan. Bring to a boil; reduce heat, and maintain at a light simmer. Add vinegar. Break eggs, and slip into water, 1 at a time, as close as possible to surface of water. Simmer 3 to 5 minutes or to desired degree of doneness. Remove with a slotted spoon. Trim edges, if desired. Spoon grits into 6 bowls; top with Pico de Gallo, eggs, and tortilla strips.