1. Using poultry shears, cut the chicken wings in half at the joints; set the drumettes in a large shallow baking dish. With the shears, snip off the strips of fat along each side of the other wing pieces. Add the pieces to the drumettes.
2. In a mini food processor, combine the chopped garlic with the jalapeños, 1/4 cup of the mustard, 2 tablespoons of the vinegar and 1/4 cup of the olive oil. Process until smooth.
3. Toss 1/2 cup of the marinade with the wings. In a small bowl, toss the chicken livers with the remaining 1/4 cup of marinade. Cover and refrigerate the wings and livers for 6 hours or for up to 2 days.
4. Light a grill. To make the glaze, in a medium bowl, whisk the remaining 3/4 cup of mustard and 1/2 cup of vinegar with the hot sauce.
5. Thread the chicken wings without crowding onto pairs of parallel 12-inch skewers; keep the drumettes and the other chicken wing pieces on separate sets of skewers. Thread the livers onto double skewers, leaving 1/2 inch between them. Season the chicken wings and livers with salt and pepper.
6. Brush the livers with the glaze and grill over a hot fire, brushing frequently with the glaze, until browned and crisp, about 2 minutes per side. Clean the grill with a wire brush if necessary, then brush the bread with olive oil and grill the slices over the hot fire until toasted. While still hot, rub the bread on both sides with the halved garlic clove. Serve immediately with the chicken livers.
7. Brush the chicken wings with the glaze and grill over a medium-hot fire, brushing generously with the glaze and turning for even browning, until cooked through and crisp, about 8 minutes per side. Serve at once.
Make Ahead: The marinated chicken wings and livers can be refrigerated for up to 2 days. The mustard-balsamic glaze can be refrigerated overnight.
Wine Recommendation: A bright, fruity Pinot Noir will flatter the sweet and tangy balsamic mustard, as well as the earthy chicken livers. Two good choices: the 1999 Kenwood Russian River Valley and the 1999 Domaine Carneros, both from California.