Preheat the oven to 425˚F, and line 2 baking sheets with parchment paper. Pre-measure all ingredients, and pull out a bowl and set up a mixer if you want to use that to mix in the eggs. I often just beat them in with a wooden spoon, but a mixer is definitely easier.
In a medium saucepan over medium heat, melt the butter in the water with the salt. Increase the heat once the butter is melted and bring the liquid to a full boil. Don't quit at a simmer.
Remove the pan from the heat and dump in the flour all at once, beating vigorously with a wooden spoon until the mixture comes together into a mass of dough that rolls around the bottom of the pan. Continue for another minute until it's very smooth, and a light crust begins to form on the bottom of the pan.
Transfer the dough to a bowl. Thoroughly whisk in the eggs one at a time with a wooden spoon and a lot of elbow grease, or with a mixer. The dough should be very smooth once all eggs are incorporated. Thoroughly beat in the Dijon, dry mustard, cheeses (except the 2 Tbsp grated cheddar for the topping), and black pepper. Get the dough as smooth as you can - it helps ensure a round, even puff during baking. Taste and adjust seasoning with additional salt and pepper if desired.
Using 1 Tablespoon dough for each gougère by either dropping them on a baking sheet from a tablespoon or by piping them. I prefer to pipe them by spooning the choux dough into a ziplock baggie, scraping the dough to the bottom and off to one side towards one of the corners using the back edge of a chef's knife, and snipping off a corner. Don't make too big a snip - it's easier to make it bigger than smaller, obvi.
Once they're all piped, round off any pointed tops with your finger. To keep the dough from sticking to your finger, first lightly dip your finger in olive oil.
Rest a jalapeño slice on each gougére, and top with a pinch of grated cheddar.
Bake for 10 minutes, and rotate the pan from bottom to top and from front to back. Bake for another 8 minutes, or until puffed and golden brown. Some chefs recommend piercing each puff with a small knife after the first 10 minutes, but I haven't found that it makes any difference.
Remove the puffs from the oven, keep on the baking sheet for one minute, and move to a cooling tray. Serve warm!