[SPARK-49747][CONNECT] Migrate connect/ files to structured logging

### What changes were proposed in this pull request?

We are moving one missing piece in SparkConnect to MDC-based logging.

### Why are the changes needed?

As part of the greater migration to structured logging.

### Does this PR introduce _any_ user-facing change?

No.

### How was this patch tested?

Compilation/existing tests.

### Was this patch authored or co-authored using generative AI tooling?

No.

Closes apache#48195 from nemanja-boric-databricks/mdc-connect.

Authored-by: Nemanja Boric <nemanja.boric@databricks.com>
Signed-off-by: Herman van Hovell <herman@databricks.com>

sql/connect/server/src/main/scala/org/apache/spark/sql/connect/execution/ExecuteThreadRunner.scala

@@ -27,7 +27,7 @@ import org.apache.commons.lang3.StringUtils
 
 import org.apache.spark.SparkSQLException
 import org.apache.spark.connect.proto
-import org.apache.spark.internal.Logging
+import org.apache.spark.internal.{Logging, LogKeys, MDC}
 import org.apache.spark.sql.connect.common.ProtoUtils
 import org.apache.spark.sql.connect.planner.SparkConnectPlanner
 import org.apache.spark.sql.connect.service.{ExecuteHolder, ExecuteSessionTag, SparkConnectService}
@@ -113,7 +113,7 @@ private[connect] class ExecuteThreadRunner(executeHolder: ExecuteHolder) extends
       } catch {
         // Need to catch throwable instead of NonFatal, because e.g. InterruptedException is fatal.
         case e: Throwable =>
-          logDebug(s"Exception in execute: $e")
+          logDebug(log"Exception in execute: ${MDC(LogKeys.EXCEPTION, e)}")
           // Always cancel all remaining execution after error.
           executeHolder.sessionHolder.session.sparkContext.cancelJobsWithTag(
             executeHolder.jobTag,
@@ -298,7 +298,7 @@ private[connect] class ExecuteThreadRunner(executeHolder: ExecuteHolder) extends
         ProtoUtils.abbreviate(request, maxLevel = 8).toString)
     } catch {
       case NonFatal(e) =>
-        logWarning("Fail to extract debug information", e)
+        logWarning(log"Fail to extract debug information: ${MDC(LogKeys.EXCEPTION, e)}")
         "UNKNOWN"
     }
   }