Heat butter in a large saucepan over medium-high heat until beginning to foam. Add chicken pieces and cook in batches, turning often, until golden brown (4-5 minutes). Remove chicken and set aside. Add onion and garlic to pan and sauté over medium heat until soft (3-4 minutes). Add mushrooms and sauté until just starting to soften (2-3 minutes). Return chicken to pan, season to taste, add beer, bring to the simmer, cover, reduce heat to low and simmer until chicken is just cooked through (30-35 minutes).
Transfer chicken to a plate and set aside. Add cream and silverbeet to pan, increase heat to medium-high and cook until liquid is reduced by half (8-10 minutes). Return chicken to pan, add tarragon and cook until warmed through (1-2 minutes). Serve immediately.
Braising in beer is common in beer-drinking nations such as Belgium. The yeasty flavour permeates the ingredients, but be careful to choose a beer that doesn’t overpower. A pale ale is best for chicken and pork, but beef is well suited to a rich dark ale.