COMBINE shallot, garlic, crabmeat, mayonnaise, chives and lemon zest in a medium bowl; season with salt. Gently fold to combine. Set aside. (Crab mixture can be made 4 hours ahead. Keep covered and chilled until use.)
POUR oil into a large saucepan fitted with a deep-fry thermometer to a depth of 6 inches. Heat oil over medium-high heat until thermometer registers 375°F.
WHILE oil is heating, whisk flour, cornstarch, baking powder and ½ teaspoon salt in a large bowl. Gradually whisk in beer, just to blend (batter will be thick).
WORKING in batches of about 4, measure 1 heaping tablespoon crab mixture per beignet, roll into a ball, and drop into batter. Using a fork, toss to coat and lift from batter, letting excess drip back into bowl. Carefully lower beignets into oil. Fry, turning occasionally, until crisp and deep golden brown, about 4 minutes, making sure to return oil to 375° between batches.
TRANSFER beignets to a paper-towel–lined plate and season with salt.