Cut off and discard top of lemongrass, leaving 6-inch stalks, then finely chop. Cook lemongrass, shallots, and garlic in oil in a large heavy pot over medium-low heat, stirring occasionally, until well browned, 12 to 15 minutes.

Add stock, reserved tomato juice, tamarind, fish sauce, chiles, lime leaves, and ginger and simmer, uncovered, 30 minutes.

While soup simmers, freeze chicken breast just until slightly firm, 20 to 30 minutes, then thinly slice crosswise.

Strain stock through a fine-mesh sieve into a large saucepan, pressing hard on and then discarding solids. Return to a simmer and stir in chicken, diced tomatoes, snow peas, and basil. Gently simmer just until chicken is cooked through, 1 to 2 minutes. Season with additional fish sauce and salt.