Preheat the oven to 400℉.
In the bowl of a food processor, place both flours, sesame seeds, and kosher salt; pulse until combined. With the processor running, slowly add oil. Add ½ cup beer and process until the dough forms a ball.
On a lightly floured surface, knead the dough 2 or 3 times. Divide dough in half. Roll half the dough into a 12-inch square. Using a fluted pastry wheel, cut dough into 2-inch squares. Poke each cracker in the center with a wooden skewer or meat thermometer. Line 2 baking sheets with parchment paper or silpats. Transfer the cracker squares to the baking sheet. Repeat with remaining dough.
Lightly brush the crackers with the remaining 1 tbs beer and sprinkle with the sea salt.
Bake until golden brown, 12 to 15 minutes. Rotate the pans halfway through the baking. Let cool completely on pans.
PUB CHEESE SPREAD:
In the bowl of a stand mixer fitted with a paddle attachment, beat Cheddar, shallot, garlic, mustard, and cayenne at low speed until combined. Slowly add beer and chives. Beat at medium-high speed until creamy, about 1 minute. Cover and refrigerate for at least 2 hours before serving. Right before you serve it, garnish the top with some snipped chives.