Preheat the oven to 400°F (200°C) and line a baking sheet with parchment paper. Set up three shallow bowls: one with crushed pretzels, one with flour, and one with the eggs and milk whisked together.

Season the chicken breasts with salt and black pepper. Dredge each piece in flour, shaking off the excess, then dip into the egg mixture and press into the crushed pretzels to coat all sides.

In a large skillet, heat the vegetable oil over medium-high heat. Sear the chicken for 3–4 minutes per side until golden and crisp.

Transfer the chicken to the prepared baking sheet and bake for 15–20 minutes, or until the thickest part reaches 165°F (74°C). Let rest a few minutes.

While the chicken bakes, melt the butter in a small saucepan over medium heat. Whisk in the flour and cook for 1–2 minutes, stirring constantly, to form a smooth roux.

Slowly whisk in the milk and cook until thickened. Reduce the heat to low, stir in the shredded cheddar until melted, then add the Dijon mustard. Season with salt and pepper to taste and stir until smooth.

Serve the pretzel-crusted chicken with warm mustard cheddar sauce spooned over the top.