Preheat oven to 350°F. Grease a 7-inch soufflé dish with the end of a 1/2 stick of butter.
Heat a skillet with cooking oil to medium heat and add jalapeños. Cook until tender, about 3-4 minutes. Add spinach and cook until wilted. Beat egg whites and cream of tartar with an electric mixer until stiff peaks form, about 5 minutes.
Melt the remaining butter in a saucepan over low heat. Once melted, stir in flour and cook until bubbly. Stir in 1 cup milk and continue stirring until the mixture thickens, about 5 minutes. Season with pepper.
In a clean bowl, whisk egg yolks for one minute. Whisk in butter, flour and milk sauce. Add spinach, jalapeños and cheese. Mix to combine and allow cheese to melt.
Stir about 1/4 of egg whites into the yolk mixture and very gently fold together. A little at a time, fold all of egg whites in and pour into greased soufflé dish. Bake for 50-60 minutes, until top is brown and puffy.