Pour oil to a depth of 1 1/2 inches in a Dutch oven; heat over medium-high heat to 350°.
Whisk together flour, cornstarch, and 1 Tbsp. salt in a large bowl; whisk in beer. Dip pumpkin wedges in batter, allowing excess batter to drip off. (Pumpkin should be very lightly coated.)
Gently lower pumpkin into hot oil, using tongs. Fry pumpkin, in 3 batches, 3 to 4 minutes or until tender inside and light brown outside, turning once. Place fried pumpkin on a wire rack in a jelly-roll pan; season with salt. Repeat procedure with sage leaves, frying 1 minute on each side.
Place peeled garlic clove on a cutting board; smash garlic, using flat side of knife, to make a paste. Whisk together garlic, yogurt, next 3 ingredients, and remaining 1/2 tsp. salt. Serve yogurt mixture with fried pumpkin and sage leaves.
