Spray a large 13 x 9 x 2" casserole dish with vegetable spray and set aside.
Heat the olive oil in a large skillet over medium high heat and add the diced bell peppers and onions, sprinkle with a pinch of kosher salt. Saute the vegetables until softened, about 3-5 minutes. Add the diced hatch chilies and continue to saute until most of the liquid has evaporated. Remove from heat. In a large bowl combine the eggs, remaining kosher salt and pepper. Whisk the eggs, then add the milk and whisk to combine.
Layer the casserole: Add the bread cubes to the casserole dish and spread into an even layer. Sprinkle with one cup each of the monterey jack and cheddar cheeses. Sprinkle the peppers in an even layer over the cheeses. Pour the egg mixture over the peppers and sprinkle on the remaining cup of monterey jack and half cup of cheddar. Lightly press the contents into the egg mixture. Cover with plastic wrap and refrigerate overnight.
Preheat the oven to 350° and bake the strata uncovered for 45-55 minutes, or until bubbly and lightly browned.