Peel the russet potatoes if desired and cut into 0.25-inch batons.
Submerge the cut potatoes in cold water and soak 30 minutes to rinse off surface starch; drain and pat very dry.
Heat the vegetable oil in a heavy pot to 325°F. Fry the potatoes in batches until pale and tender but not colored, 5–6 minutes; drain and rest 10 minutes.
Increase oil temperature to 375°F. Fry the blanched potatoes again until golden and very crisp, 2–4 minutes; drain well.
Meanwhile, melt the unsalted butter in a saucepan over medium heat. Whisk in the all-purpose flour and cook, whisking, until foamy and blond, 1–2 minutes.
Gradually whisk in the whole milk. Bring to a gentle simmer and cook, stirring, until smooth and thick enough to coat a spoon, 3–5 minutes.
Remove from heat and whisk in the cheddar cheese, Dijon mustard, cayenne pepper, and black pepper until the sauce is glossy and smooth; keep warm.
Taste and season both the hot fries and the cheese sauce with the kosher salt to taste; toss the fries and whisk the sauce to distribute.
Pile the fries on a warm platter, spoon the cheese sauce evenly over the top, and finish with scallions and pickled jalapeño slices, if using. Serve immediately.