Heat a medium saucepan over medium heat. Add oil to pan; swirl to coat. Add shallots; cook 5 minutes or until shallots begin to brown, stirring frequently. Add garlic; cook 1 minute or until fragrant. Stir in stock; cook 30 seconds. Transfer stock mixture to a blender; process until smooth.
Return stock mixture to pan; add potatoes, stirring with a whisk until combined. Stir in 1/4 cup basil, thyme, and 1 tablespoon chives; bring to a boil. Cook 1 minute. Remove from heat; discard thyme. Stir in half-and-half, cheese, and pepper. Sprinkle with remaining 1/4 cup basil and remaining 1 tablespoon chives.
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