Preheat the oven to 375°F. Place garlic in a small ovenproof skillet or baking pan and toss with 1/4 teaspoon of the oil. Roast until cloves are soft and browned, about 25 minutes. Cool, then peel garlic. Combine garlic, eggs, butter, 1/2 teaspoon of the salt and pepper in a blender or food processor and purée. Pour mixture into a bowl and stir in matzoh meal and 3 tablespoons water. Cover and refrigerate for 30 minutes.
Meanwhile, heat remaining 1/2 teaspoon oil in a large saucepan over medium-high heat. Add onion, carrots, celery and jalapeño and cook, stirring occasionally, until onion softens, about 3 minutes. Add broth and remaining 1/4 teaspoon salt and bring to a boil.
With damp hands, form matzoh mixture into 16 balls, each about the size of a ping-pong ball, and place on a plate. When broth boils, add balls, lower heat and simmer gently until matzoh balls are cooked through, about 25 minutes. Discard jalapeño. Ladle matzoh balls, broth and vegetables into bowls.