Cut open the roasted green chiles and scrape out the seeds. Then chop the green chiles and shallots.
In a medium pot over medium heat, melt the butter and add shallots and chiles. Cook, stirring regularly, until the shallots are softened, 3 to 4 minutes.
Stir the flour into the pot and cook for a minute or two until the mixture forms a paste.
Slowly drizzle in the milk, stirring constantly to avoid lumps. A whisk helps, but you can also just use a spoon.
Once the milk is added, stir in the grated cheese. When the cheese is melted, season with a pinch of salt and pepper.
Serve queso immediately, while hot, with tortilla chips and garnish with a little extra green chile on top.
Notes
If you cannot find fresh roasted Hatch chiles in your area, you can substitute one 5-ounce can of chopped green chiles, either mild or hot, depending on your preference.
Roast your own Hatch chiles: Place the chiles on a baking sheet: Arrange the chiles in a single layer on an aluminum foil-lined baking sheet. Roast the chiles on one side: Place the chiles under the broiler and roast until the skin is charred and blistered, about 3 to 5 minutes.