Heat oven to 450 degrees F and arrange racks on bottom third and upper third of oven. Line two baking sheets with parchment paper or silicone mats.

In a medium skillet over medium-low heat, combine 1 tablespoon butter and sliced shallots and cook, stirring often, until shallots are caramelized to a crisp golden brown.

In a medium saucepan over medium heat, combine remaining 6 tablespoons butter, water, mustard, salt and pepper. Melt butter and bring to a simmer. Add flour and immediately stir with a wooden spoon or rubber spatula. Continue stirring 5 minutes until a ball of dough forms and a thin film of flour develops on the bottom and sides of the pan.

Transfer dough to bowl of a stand mixer fitted with paddle attachment. Stir on medium speed 1 minute to release some steam from the dough. Stir in 1 egg at a time, beating well at medium speed after each addition until just incorporated. Dough will be clumpy at first, but by the final egg it should be smooth, glossy and just barely pourable.

Use a spoon, scoop or a piping bag with a 1/2-inch round tip to form dollops of batter onto prepared baking sheets. Dollops should be about 1 1/2 inches wide and spaced at least 1 inch apart.

Place one baking sheet on each rack and bake 10 minutes until gougeres are puffy. Rotate pans (place upper baking sheet on lower rack, and lower baking sheet on upper rack) and reduce oven temperature to 350 degrees F. Bake another 15 to 20 minutes until gougeres are puffy, golden brown and just baked through (to test, remove one from the oven, pull it apart and if the inside is moist but not gooey, they're done. Otherwise, let them bake a little longer).

Serve gougeres warm or at room temperature.