Cut open the roasted green chiles and scrape out the seeds. Then chop the green chiles and shallots.: The first sensory cue is the fragrant, slightly smoky scent released as you slice into the roasted Hatch chiles , and the shallots should smell sharp and slightly sweet as you chop them. Doing this prep ensures even distribution of chile pieces, so each dip bite has balanced heat. A common mistake is leaving too many seeds, which can make the queso hotter than intended; scrape gently and check under a light. Keep your knife steady and use a cutting board with a groove if possible to catch juices.
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.: At this stage you should hear a quiet sizzle as the butter foams and the chopped shallots begin to soften, releasing a sweet, caramelizing aroma that blends with the chile's smokiness. This softening mellows the shallots and helps the chiles’ flavor infuse the fat, which is essential for a cohesive taste. Avoid browning the shallots heavily, which would introduce bitterness; maintain medium heat and stir often so nothing sticks to the pot bottom. If your heat is too high, lower it and continue stirring until softened.
Stir the flour into the pot and cook for a minute or two until the mixture forms a paste.: When you add the all purpose flour , it should immediately start to coat the softened aromatics, and the paste will look glossy and smell faintly nutty after a minute or two. Cooking the flour briefly removes the raw taste and creates the backbone for a silky sauce. A mistake here is not cooking long enough, leaving a pasty floury note in the finished queso; cook until the mixture loses its raw smell but before it browns. Stir continuously so the roux is uniform and smooth.
Slowly drizzle in the milk, stirring constantly to avoid lumps. A whisk helps, but you can also just use a spoon.: As you add the milk , do it in a slow, steady stream while stirring vigorously so the roux incorporates without forming lumps; you will hear a gentle bubbling as the liquid warms and thickens. The sauce should transform into a glossy, velvety béchamel like base. If lumps form, switch to a whisk and work them out immediately, or pass the sauce through a sieve if needed. Keep the heat moderate so the milk does not scorch on the pot bottom.
Once the milk is added, stir in the grated cheese. When the cheese is melted, season with a pinch of salt and pepper.: As you add the shredded Monterey Jack cheese , the mixture will shift from creamy to luxuriously melted, with a slightly elastic sheen as the cheese integrates; the aroma becomes rich and dairy forward. Stir gently and patiently until the cheese disappears into a smooth sauce. Taste before adding more salt , because the cheese contributes seasoning. A frequent error is turning the heat too high while melting the cheese, causing it to seize into oily clumps; keep heat low and stir constantly for a silky result.
Serve queso immediately, while hot, with tortilla chips and garnish with a little extra green chile on top.: The final presentation should steam gently and glisten, inviting immediate dipping with crisp tortilla chips . Serve right away so the texture is warm and velvety; as it cools it will thicken. If you must hold it, keep it over very low heat and stir occasionally. A common misstep is leaving the queso to sit too long, letting it firm up and separate; warm it gently and whisk to re-emulsify if necessary before serving.