Push the meat to one side of the pan and add the onion, garlic, jalapeño, chili powder and cumin to the pot. Cook until the veggies soften, 5-6 minutes, and season with salt and freshly ground black pepper.
Add the tomato paste to the pot and cook for about 1 minute, until it smells sweet and has caramelized slightly. Add the hot sauce, Worcestershire and beer or stock to the pot. Bring everything up to a bubble, then reduce the heat and simmer the chili until thickened, about 10 minutes.
While the chili is simmering, place a medium-size pot over medium-high heat and melt the butter. Sprinkle the flour over the melted butter and cook it for about a minute. Stir the mustard into the butter-flour mixture, then whisk in the milk and stock. Continue whisking the sauce until it has thickened up, then remove the pot from the heat and stir in the cheese until melted. Season the cheese sauce with salt and freshly ground black pepper and reserve.
In a large salad bowl, whisk together 1/4 cup EVOO along with the red wine vinegar, salt and freshly ground black pepper. Add the lettuce and cucumber to the bowl and toss to combine. To serve, divide the fries between four plates and top each one with some chili-cheese sauce and some salad alongside.