botfather: /start <bot> opens that bot's menu

The "Manage Bot" button on a bot's profile deep-links to @BotFather with
start=<bot username>. parseBotCommand dropped the argument, so /start <bot>
just replied with the generic greeting instead of the per-bot menu.

Route "/start <arg>" to the bot's "What do you want to do?" screen (same as
/mybots then tapping the bot) when <arg> names one of the sender's own bots by
username or id; empty or unknown args keep the greeting.

internal/app/bots/botfather.go

@@ -274,6 +274,13 @@ func (s *Service) handleBotFather(ctx context.Context, userID int64, msg domain.
 	if cmd, ok := parseBotCommand(text); ok {
 		inValueStep := found && state.Step == botFatherStepValue
 		if !inValueStep || botFatherGlobalCommands[cmd] {
+			// "/start <bot>" (the "Manage Bot" deep link) jumps straight to that
+			// bot's menu, like /mybots then tapping the bot.
+			if cmd == "start" {
+				if arg := botCommandArg(text); arg != "" {
+					return s.handleBotFatherStart(ctx, userID, arg)
+				}
+			}
 			return s.handleBotFatherCommand(ctx, userID, cmd)
 		}
 	}
@@ -368,6 +375,41 @@ func (s *Service) stepPrompt(state domain.BotChatState) botReply {
 	}
 }
 
+// handleBotFatherStart answers "/start <arg>". When <arg> names one of the
+// user's own bots (by username or numeric id) it opens that bot's menu - the
+// same "What do you want to do?" screen as /mybots then tapping the bot, which
+// is what the "Manage Bot" button on a bot's profile links to. An empty or
+// unknown arg falls back to the plain greeting.
+func (s *Service) handleBotFatherStart(ctx context.Context, userID int64, arg string) botReply {
+	_ = s.bots.DeleteBotChatState(ctx, domain.BotFatherUserID, userID)
+	want := strings.ToLower(strings.TrimPrefix(strings.TrimSpace(arg), "@"))
+	if want == "" {
+		return botReply{Text: botFatherHelpText}
+	}
+	owned, err := s.ownedBots(ctx, userID)
+	if err != nil {
+		s.log.Error("botfather: list bots for start payload", zap.Int64("user_id", userID), zap.Error(err))
+		return internalReply()
+	}
+	for _, b := range owned {
+		if strings.EqualFold(b.user.Username, want) || strconv.FormatInt(b.user.ID, 10) == want {
+			state := domain.BotChatState{
+				BotUserID: domain.BotFatherUserID,
+				UserID:    userID,
+				Command:   mybotsCommand,
+				Step:      mybotsStepMenu,
+				Draft:     map[string]string{},
+			}
+			reply := s.myBotsBotMenu(&state, b)
+			if !s.saveMyBotsState(ctx, state) {
+				return internalReply()
+			}
+			return reply
+		}
+	}
+	return botReply{Text: botFatherHelpText}
+}
+
 func (s *Service) handleBotFatherCommand(ctx context.Context, userID int64, cmd string) botReply {
 	switch cmd {
 	case "start", "help":
@@ -1157,3 +1199,16 @@ func parseBotCommand(text string) (string, bool) {
 	}
 	return strings.ToLower(cmd), true
 }
+
+// botCommandArg returns the trimmed argument after a leading "/cmd", e.g.
+// "/start my_bot" -> "my_bot". Empty when there is no argument.
+func botCommandArg(text string) string {
+	text = strings.TrimSpace(text)
+	if !strings.HasPrefix(text, "/") {
+		return ""
+	}
+	if i := strings.IndexAny(text, " \t\n"); i >= 0 {
+		return strings.TrimSpace(text[i+1:])
+	}
+	return ""
+}